text: a measure would be very useful, for example in order to se- ror itself, is used to calculate the pairwise consistency. We lect a proper alignment model for a given dataset or to select observe that the consistency calculated by our selection pro- which samples to annotate in an Active Learning scheme. cess is very close to the one calculated based on the ground Here, we show how the mirror error can be used for select- truth. We can further conclude that: • ing difﬁcult samples in the problem of face alignment. In the difﬁculty of samples is shared by the different order to do so we apply several methods (IFA, SDM, GN- methods that we have examined. DPM, RCPR) on the test images of the 300W and get the • the difﬁcult samples selected by the mirror error show detection results. Then we sort the normalized mirror error high consistency across different approaches. e in descending order and select the ﬁrst M samples as m being the most difﬁcult ones. We denote this set as S . In order to evaluate whether the samples that weemhave 3.2. Feedback on cascaded face alignment selected in this way are truly ’difﬁcult’ we measure the sim- In recent years cascaded methods like SDM [22], IFA ilarity between the set containing those M selected samples [2], CFAN [25] and RCPR [4] have shown promising re- and the set S that contains the M samples that have the ea sults in face alignment. Although they differ in terms of largest alignment error e for each method. We use a mea- a the regressor and the features that they use in each iteration sure that we call consistency which we deﬁne as the fraction they all follow the same strategy. The methods start from of the common samples between the two sets, that is one or several initializations of the face shape, that are often | ∩ | calculated from the face bounding box, and then iteratively S S ρ = 1 2 (3) reﬁne the estimation of the face shape by applying at each M iteration a regressor that estimates the udpate of the shape. These methods are intrinsically sensitive to the initialization | ∩ | where S S is the size of the intersection of S and [4, 25] . As stated in [23], only initializations that are in a 1 2 1 S . For each method i, we calculate two sets each contain- range of the optimal shape can converge to the correct so- 2 ing M samples, i.e., Si and Si . We set the value of M lution. To address this problem, [5] proposed to use several em ea to 150. The chance rate is M , where M is the number of random initializations and give the ﬁnal estimate as the me- N selected and N is the size of the dataset - in our case is dian of the solutions to which they convergence. However, ≈ 150 0.22. having several randomly generated initializations does not 689 The pairwise consistency rate matrix of Si and Si is guarantee that the correct solution is reached. The ’smart shown in Fig. 8a, where in a certain row we sehmow the ecaon- restart’ proposed in [4] has improved the results to a certain sistency between the Si of a certain method with the Si degree. The scheme starts from different initializations and of all methods, includinegmthe method itself. Note that the deia- apply only 10% of the cascade. Then, the variance between agonal does not contain ones, since Si are the M samples the predictions is checked. If the variance is below a certain with the highest mirror error and Si emthe M samples with threshold, the remaining 90% of the cascade is applied as the highest alignment error. As it ceaan be seen, the consis- usual. Otherwise the process is restarted with a different set tency between the two sets of samples for a speciﬁc method of initializations. (i.e., the diagonal values) are all above 0.7 - the highest is Here, we propose to use the mirror error as a feedback 0.81 for RCPR. More interestingly, the consistency across to close this open cascaded system. More speciﬁcally, for different methods, i.e., the M samples selected according a given test image we ﬁrst create its mirror image. Then to e for a method in a certain row and the M samples se- we apply the RCPR model on the original test image and a lected according to e in a certain column is high, with val- the mirror image and calculate the mirror error. If the mir- m ues ranging from 0.56 to 0.68. This shows that the samples ror error is above a threshold we restart the process using that we have selected are truly ’difﬁcult’, not only for the different initializations, otherwise we keep the detection re- method employed in the selection process but also for the sults. This procedure can be applied until the mirror error is other face alignment methods. In other words this shows below a threshold, or until a maximum number of iterations that the methods that we have examined have difﬁculties M is reached. In contrast to the original RCPR method that with the same images. keeps only the results from the last set of initializations, we Second, we evaluate the consistency across different ap- keep the one that has the smallest mirror error. This makes proaches, i.e., the consistency of ’difﬁcult’ samples found sense since new random initializations do not necessarily by different approaches. Thus, we calculate the pair- lead to better results than past initializations. wise consistency of Si of those methods as shown in First we evaluate the effectiveness of our feedback Fig. 8b. The resulting vemalues are clearly much higher than scheme. Ideally, the restart will be initiated only when the the chance value of 0.22. In Fig. 8c we depict the ’opti- current initialization is unable to lead to a good solution. mal’ case where the ground truth, that is the alignment er- Treating it as a two class classiﬁcation problem we report ##LTLine## ##LTLine## ##LTLine##

text: a

text: measure

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text: truth.

text: We

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text: samples

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text: problem

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text: regressor

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text: of

text: the

text: common

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text: the

text: two

text: sets,

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text: is

text: one

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text: the

text: face

text: shape,

text: that

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text: often

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text: the

text: face

text: bounding

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text: and

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text: S

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text: ρ

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text: the

text: estimation

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text: regressor

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text: udpate

text: of

text: the

text: shape.

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text: the

text: initialization

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text: S

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text: the

text: size

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text: contain-

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text: optimal

text: shape

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text: converge

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text: correct

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text: ing

text: M

text: samples,

text: i.e.,

text: Si

text: and

text: Si

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text: set

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text: value

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text: M

text: lution.

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text: solutions

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text: convergence.

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text: not

text: The

text: pairwise

text: consistency

text: rate

text: matrix

text: of

text: Si

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text: guarantee

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text: the

text: correct

text: solution

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text: reached.

text: The

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text: that

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text: highest

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text: it

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text: be

text: seen,

text: the

text: consis-

text: usual.

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text: the

text: process

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text: restarted

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text: a

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text: method

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text: (i.e.,

text: the

text: diagonal

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text: are

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text: highest

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text: we

text: propose

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text: the

text: mirror

text: error

text: as

text: a

text: feedback

text: for

text: RCPR.

text: More

text: interestingly,

text: the

text: consistency

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text: M

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text: image

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text: ﬁrst

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text: for

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text: method

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text: M

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text: we

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text: RCPR

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text: original

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text: column

text: is

text: high,

text: with

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text: the

text: mirror

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text: the

text: mirror

text: error.

text: If

text: the

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text: m

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text: to

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text: samples

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text: not

text: only

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text: different

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text: otherwise

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text: the

text: detection

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text: method

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text: the

text: selection

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text: sults.

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text: procedure

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text: applied

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text: the

text: mirror

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text: In

text: other

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text: threshold,

text: or

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text: number

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text: the

text: methods

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text: difﬁculties

text: M

text: is

text: reached.

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text: the

text: original

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text: that

text: with

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text: same

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text: last

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text: consistency

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text: report

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 with − ∂G 1(w) − − (cid:48) = iγ0δ(r r ). ∂w The multiplier T came from the functional trace in time, and now Tr contains only spatial integration. By substituting − the expression for the Green function (A15) into Ω = Γ/TV , one can see that Schwinger phase Φ goes away and after the Fourier tran(cid:90)sformati(cid:90)on we get f(cid:20)or the energy density (cid:21) ∞ (cid:90) dw (cid:90) d2k 1 − { − − · } − Ω = i tr iγ0wG¯(w, k) + [ i (w (cid:15) σ )γ0 v (k γ) G¯(w; k) 1] −∞ 2π (2π)2 2 Z Z F ∞ − dw d2k { · } = tr [(w + (cid:15) σ )γ0 + v (k γ)]G¯(w; k) + i . (C3) −∞ 4π (2π)2 Z Z F By making use of the explicit form of the propagator, we calculate the following integrals which contribute to the energy density: (cid:90) (cid:88) | | ∞ − d2k i eB P− + P θ(n 1/2) (cid:90) γ0G¯ (w; k) = γ0W(cid:88) M+− , (C4) (2π)2 s 2π n(cid:15)2 n=0 B | | ∞ d2k · i eB n(cid:15)2 v (k γ)G¯ (w; k) = M −B , (C5) (2π)2 F s 2π n(cid:15)2 n=0 B − where in strained case we use the eﬀective magnetic ﬁeld B = B⊥ s s B in the Landau energy (cid:15) and projectors 0 12 5 B P±. By dropping an inﬁnite divergent term independent of the physical parameters and normalizing Ω by subtracting its value at mµ(cid:90) = dµ = (cid:15)Z(cid:88)= (cid:20)0, we obtain the following expression: (cid:21) | | ∞ ∞ − − − i eB dw (w + (cid:15) σ )γ0W(P− + P θ(n 1/2)) + n(cid:15)2 − w2(P− + P θ(n 1/2)) + n(cid:15)2 Ω = tr Z Z M − + B + − B . (C6) 4π −∞ 2π n(cid:15)2 w2 n(cid:15)2 n=0 B B After integrating this expression over frequency and taking trace, we ﬁnally arrive at Eq.(43). 1 X. Li, X. Wang, L. Zhang, S. Lee, and H. Dai, Science 319, 1229 (2008). 2 T. Low and F. Guinea, Nano Letters 10, 3551 (2010). 3 J. Lu, A.C. Neto, and K.P. Loh, Nature Communications 3, 823 (2012). 4 H. Suzuura and T. Ando, Phys. Rev. B 65, 235412 (2002). 5 J.L. Manes, Phys. Rev. B 76, 045430 (2007). 6 S.V. Morozov, K.S. Novoselov, M.I. Katsnelson, F. Schedin, D. Jiang, and A.K. Geim, Phys. Rev. Lett. 97, 016801 (2006). 7 T.O. Wehling, A.V. Balatsky, A.M. Tsvelik, M.I. Katsnelson, and A.I. Lichtenstein, Europhys. Lett. 84, 17003 (2008). 8 M.A. Vozmediano, M. Katsnelson, and F. Guinea, Physics Reports 496, 109 (2010). 9 M.I. Katsnelson, Graphene: Carbon in two dimensions (Cambridge University Press, Cambridge, 2012). 10 F. Guinea, M.I. Katsnelson, and A.K. Geim, Nature Physics 6, 30 (2010). 11 F. Guinea, A.K. Geim, M.I. Katsnelson, and K.S. Novoselov, Phys. Rev. B 81, 035408 (2010). 12 K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jaing, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov, Nature 438, 197 (2005); Y. Zhang, Y.-W. Tan, H.L. Sto¨rmer, and P. Kim, Nature 438, 201 (2005). 13 V.P. Gusynin and S.G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005). 14 N.M.R. Peres, F. Guinea, and A.H. Castro Neto, Phys. Rev. B 73, 1254111 (2006). 15 Y. Zhang, Z. Jiang, J.P. Small, M.S. Purewal, Y.-W. Tan, M. Fazlollahi, J.D. Chudow, J.A. Jaszczak, H.L. Stormer, and P. Kim, Phys. Rev. Lett. 96, 136806 (2006). 16 Z. Jiang, Y. Zhang, H.L. Stormer, and P. Kim, Phys. Rev. Lett. 99, 106802 (2007). 17 N. Levy, S.A. Burke, K.L. Meaker, M. Panlasigui, A.Zettl, F. Guinea, A.H. Castro Neto, and M.F. Crommie, Science 329, 544 (2010). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: with

text: ∂G

text: r

text: ∂w

text: The

text: multiplier

text: T

text: came

text: from

text: the

text: functional

text: trace

text: in

text: time,

text: and

text: now

text: Tr

text: contains

text: only

text: spatial

text: integration.

text: By

text: substituting

text: the

text: expression

text: for

text: the

text: Green

text: function

text: into

text: Ω

text: Γ/TV

text: one

text: can

text: see

text: that

text: Schwinger

text: phase

text: Φ

text: goes

text: away

text: and

text: after

text: the

text: Fourier

label: equation

text: we

text: get

text: the

text: energy

text: density

text: dw

text: Ω

text: i

text: tr

text: k)

text: i

text: (w

text: σ

text: v

text: (k

text: γ)

text: G¯(w;

text: k)

text: 2π

text: Z

text: Z

text: F

text: dw

text: tr

text: [(w

text: σ

text: v

text: (k

text: γ)]G¯(w;

text: k)

text: i

text: 4π

text: Z

text: Z

text: F

text: By

text: making

text: use

text: of

text: the

text: explicit

text: form

text: of

text: the

text: propagator,

text: we

text: calculate

text: the

text: following

text: integrals

text: which

text: contribute

text: to

text: the

text: energy

text: density:

text: i

text: eB

text: P−

text: P

text: θ(n

text: γ0G¯

text: (w;

text: k)

text: M+−

text: s

text: 2π

text: B

text: i

text: eB

text: v

text: (k

text: γ)G¯

text: (w;

text: k)

text: M

text: −B

text: F

text: s

text: 2π

text: B

text: where

text: in

text: strained

text: case

text: we

text: use

text: the

text: eﬀective

text: magnetic

text: ﬁeld

text: B

text: B⊥

text: s

text: s

text: B

text: in

text: the

text: Landau

text: energy

text: and

text: projectors

text: B

text: P±.

text: By

text: dropping

text: an

text: inﬁnite

text: divergent

text: term

text: independent

text: of

text: the

text: physical

text: parameters

text: and

text: normalizing

text: Ω

text: by

text: subtracting

text: its

text: value

text: at

text: dµ

text: we

text: obtain

text: the

text: following

text: expression:

text: i

text: eB

text: dw

text: (w

text: σ

text: P

text: θ(n

text: P

text: θ(n

text: Ω

text: tr

text: Z

text: Z

text: M

text: B

text: B

text: 4π

text: 2π

text: B

text: B

text: After

text: integrating

text: this

text: expression

text: over

text: frequency

text: and

text: taking

text: trace,

text: we

text: ﬁnally

text: arrive

text: at

text: X.

text: Li,

text: X.

text: Wang,

text: L.

text: Zhang,

text: S.

text: Lee,

text: and

text: H.

text: Dai,

text: Science

text: T.

text: Low

text: and

text: F.

text: Guinea,

text: Nano

text: Letters

text: J.

text: Lu,

text: A.C.

text: Neto,

text: and

text: K.P.

text: Loh,

text: Nature

text: Communications

text: H.

text: Suzuura

text: and

text: T.

text: Ando,

text: Phys.

text: Rev.

text: B

text: J.L.

text: Manes,

text: Phys.

text: Rev.

text: B

text: S.V.

text: Morozov,

text: K.S.

text: Novoselov,

text: M.I.

text: Katsnelson,

text: F.

text: Schedin,

text: D.

text: Jiang,

text: and

text: A.K.

text: Geim,

text: Phys.

text: Rev.

text: Lett.

text: T.O.

text: Wehling,

text: A.V.

text: Balatsky,

text: A.M.

text: Tsvelik,

text: M.I.

text: Katsnelson,

text: and

text: A.I.

text: Lichtenstein,

text: Europhys.

text: Lett.

text: M.A.

text: Vozmediano,

text: M.

text: Katsnelson,

text: and

text: F.

text: Guinea,

text: Physics

text: Reports

text: M.I.

text: Katsnelson,

text: Graphene:

text: Carbon

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text: two

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text: (Cambridge

text: University

text: Press,

text: Cambridge,

text: F.

text: Guinea,

text: M.I.

text: Katsnelson,

text: and

text: A.K.

text: Geim,

text: Nature

text: Physics

text: F.

text: Guinea,

text: A.K.

text: Geim,

text: M.I.

text: Katsnelson,

text: and

text: K.S.

text: Novoselov,

text: Phys.

text: Rev.

text: B

text: K.S.

text: Novoselov,

text: A.K.

text: Geim,

text: S.V.

text: Morozov,

text: D.

text: Jaing,

text: M.I.

text: Katsnelson,

text: I.V.

text: Grigorieva,

text: S.V.

text: Dubonos,

text: and

text: A.A.

text: Firsov,

text: Nature

text: Y.

text: Zhang,

text: Y.-W.

text: Tan,

text: H.L.

text: Sto¨rmer,

text: and

text: P.

text: Kim,

text: Nature

text: V.P.

text: Gusynin

text: and

text: S.G.

text: Sharapov,

text: Phys.

text: Rev.

text: Lett.

text: N.M.R.

text: Peres,

text: F.

text: Guinea,

text: and

text: A.H.

text: Castro

text: Neto,

text: Phys.

text: Rev.

text: B

text: Y.

text: Zhang,

text: Z.

text: Jiang,

text: J.P.

text: Small,

text: M.S.

text: Purewal,

text: Y.-W.

text: Tan,

text: M.

text: Fazlollahi,

text: J.D.

text: Chudow,

text: J.A.

text: Jaszczak,

text: H.L.

text: Stormer,

text: and

text: P.

text: Kim,

text: Phys.

text: Rev.

text: Lett.

text: Z.

text: Jiang,

text: Y.

text: Zhang,

text: H.L.

text: Stormer,

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text: P.

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text: Rev.

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text: N.

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text: K.L.

text: Meaker,

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text: F.

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text: A.H.

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text: Catalan Pairs and Fishburn Triples V´ıt Jel´ınek∗ May 27, 2020 Abstract Disanto, Ferrari, Pinzani and Rinaldi have introduced the concept of Catalan pair, which is a pair of partial orders (S, R) satisfying certain axioms. They have shown that Catalan pairs provide a natural description of objects belonging to several classes enumerated by Catalan numbers. In this paper, we ﬁrst introduce another axiomatic structure (T, R), which we call the Catalan pair of type 2, which describes certain Catalan objects that do not seem to have an easy interpretation in terms of the original Catalan pairs. We then introduce Fishburn triples, which are relational structures obtained as a direct common generalization of the two types of Catalan pairs. Fishburn triples encode, in a natural way, the structure of objects enumerated by the Fishburn numbers, such as interval orders or Fishburn matrices. This connection between Catalan objects and Fishburn objects allows us to associate known statistics on Catalan objects with analogous statistics of Fishburn objects. As our main result, we then show that several known equidistribution results on Catalan statistics can be generalized to analogous results for Fishburn statistics. (cid:0) (cid:1) 1 Introduction The Catalan numbers C = 1 2n are one of the most ubiquitous number sequences in enumerative n n+1 n combinatorics. As of May 2013, Stanley’s Catalan Addendum [35] includes over 200 examples of classes of combinatorial objects enumerated by Catalan numbers, and more examples are constantly being discovered. The Fishburn numbers F (sequence A022493 in OEIS [30]) are another example of a counting sequence n that arises in several seemingly unrelated contexts. The ﬁrst widely studied combinatorial class enumerated by Fishburn numbers is the class of interval orders, also known as (2+2)-free posets. Their study was pioneered by Fishburn [16, 17, 18]. Later, more objects counted by Fishburn numbers were identiﬁed, including non-neighbor-nesting matchings [36], ascent sequences [3], Fishburn matrices [12, 14], several classes of pattern-avoiding permutations [3, 31] or pattern-avoiding matchings [26]. It has been observed by several authors that various Fishburn classes contain subclasses enumerated by Catalan numbers [4, 7, 15, 24]. For instance, the Fishburn class of (2+2)-free posets contains the subclasses of (2+2, 3+1)-free posets and (2+2, N)-free posets, which are both enumerated by Catalan numbers. The aim of this paper is to describe a close relationship between Catalan and Fishburn classes, of which the (cid:0) (cid:1) above-mentioned inclusions are a direct consequence. At ﬁrst sight, Catalan numbers and Fishburn numbers do not seem to have much in common. The Catalan numbers can be expressed a simple formula C = 1 2n , they have a simply exponen√tial asymp- − n n+1 n (cid:80) (cid:81) − − totic growth C = Θ(4nn 3/2), and admit an algebraic generating function, namely (1 1 4x)/2. n √ In contrast, no simple formula for the Fishburn numbers F is known, their growth is superexponential n − − (F = Θ(n!(6/π)n n) as shown by Zagier [38]), and their generating function ≥ n (1 (1 x)k), n n 0 k=1 derived by Zagier [38], is not even D-ﬁnite [3]. ∗ Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Malostransk´e n´amˇest´ı 25, 118 00, Prague, jelinek@iuuk.mff.cuni.cz. Supported by project CE-ITI (GBP202/12/G061) of the Czech Science Foundation. 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Catalan

label: title

text: Pairs

text: and

text: Fishburn

text: Triples

text: V´ıt

label: author

text: Jel´ınek∗

text: May

text: Abstract

text: Disanto,

label: abstract

text: Ferrari,

text: Pinzani

text: and

text: Rinaldi

text: have

text: introduced

text: the

text: concept

text: of

text: Catalan

text: pair,

text: which

text: is

text: a

text: pair

text: of

text: partial

text: orders

text: (S,

text: R)

text: satisfying

text: certain

text: axioms.

text: They

text: have

text: shown

text: that

text: Catalan

text: pairs

text: provide

text: a

text: natural

text: description

text: of

text: objects

text: belonging

text: to

text: several

text: classes

text: enumerated

text: by

text: Catalan

text: numbers.

text: In

text: this

text: paper,

text: we

text: ﬁrst

text: introduce

text: another

text: axiomatic

text: structure

text: (T,

text: R),

text: which

text: we

text: call

text: the

text: Catalan

text: pair

text: of

text: type

text: which

text: describes

text: certain

text: Catalan

text: objects

text: that

text: do

text: not

text: seem

text: to

text: have

text: an

text: easy

text: interpretation

text: in

text: terms

text: of

text: the

text: original

text: Catalan

text: pairs.

text: We

text: then

text: introduce

text: Fishburn

text: triples,

text: which

text: are

text: relational

text: structures

text: obtained

text: as

text: a

text: direct

text: common

text: generalization

text: of

text: the

text: two

text: types

text: of

text: Catalan

text: pairs.

text: Fishburn

text: triples

text: encode,

text: in

text: a

text: natural

text: way,

text: the

text: structure

text: of

text: objects

text: enumerated

text: by

text: the

text: Fishburn

text: numbers,

text: such

text: as

text: interval

text: orders

text: or

text: Fishburn

text: matrices.

text: This

text: connection

text: between

text: Catalan

text: objects

text: and

text: Fishburn

text: objects

text: allows

text: us

text: to

text: associate

text: known

text: statistics

text: on

text: Catalan

text: objects

text: with

text: analogous

text: statistics

text: of

text: Fishburn

text: objects.

text: As

text: our

text: main

text: result,

text: we

text: then

text: show

text: that

text: several

text: known

text: equidistribution

text: results

text: on

text: Catalan

text: statistics

text: can

text: be

text: generalized

text: to

text: analogous

text: results

text: for

text: Fishburn

text: statistics.

text: Introduction

text: The

text: Catalan

text: numbers

text: C

text: are

text: one

text: of

text: the

text: most

text: ubiquitous

text: number

text: sequences

text: in

text: enumerative

text: n

text: n

text: combinatorics.

text: As

text: of

text: May

text: Stanley’s

text: Catalan

text: Addendum

text: includes

text: over

text: examples

text: of

text: classes

text: of

text: combinatorial

text: objects

text: enumerated

text: by

text: Catalan

text: numbers,

text: and

text: more

text: examples

text: are

text: constantly

text: being

text: discovered.

text: The

text: Fishburn

text: numbers

text: F

text: (sequence

text: in

text: OEIS

text: are

text: another

text: example

text: of

text: a

text: counting

text: sequence

text: n

text: that

text: arises

text: in

text: several

text: seemingly

text: unrelated

text: contexts.

text: The

text: ﬁrst

text: widely

text: studied

text: combinatorial

text: class

text: enumerated

text: by

text: Fishburn

text: numbers

text: is

text: the

text: class

text: of

text: interval

text: orders,

text: also

text: known

text: as

text: posets.

text: Their

text: study

text: was

text: pioneered

text: by

text: Fishburn

text: Later,

text: more

text: objects

text: counted

text: by

text: Fishburn

text: numbers

text: were

text: identiﬁed,

text: including

text: non-neighbor-nesting

text: matchings

text: ascent

text: sequences

text: Fishburn

text: matrices

text: several

text: classes

text: of

text: pattern-avoiding

text: permutations

text: or

text: pattern-avoiding

text: matchings

text: It

text: has

text: been

text: observed

text: by

text: several

text: authors

text: that

text: various

text: Fishburn

text: classes

text: contain

text: subclasses

text: enumerated

text: by

text: Catalan

text: numbers

text: For

text: instance,

text: the

text: Fishburn

text: class

text: of

text: posets

text: contains

text: the

text: subclasses

text: of

text: posets

text: and

text: N)-free

text: posets,

text: which

text: are

text: both

text: enumerated

text: by

text: Catalan

text: numbers.

text: The

text: aim

text: of

text: this

text: paper

text: is

text: to

text: describe

text: a

text: close

text: relationship

text: between

text: Catalan

text: and

text: Fishburn

text: classes,

text: of

text: which

text: the

text: above-mentioned

text: inclusions

text: are

text: a

text: direct

text: consequence.

text: At

text: ﬁrst

text: sight,

text: Catalan

text: numbers

text: and

text: Fishburn

text: numbers

text: do

text: not

text: seem

text: to

text: have

text: much

text: in

text: common.

text: The

text: Catalan

text: numbers

text: can

text: be

text: expressed

text: a

text: simple

text: formula

text: C

text: they

text: have

text: a

text: simply

text: exponen√tial

text: asymp-

text: n

text: n

text: totic

text: growth

text: C

text: and

text: admit

text: an

text: algebraic

text: generating

text: function,

text: namely

text: n

text: In

text: contrast,

text: no

text: simple

text: formula

text: for

text: the

text: Fishburn

text: numbers

text: F

text: is

text: known,

text: their

text: growth

text: is

text: superexponential

text: n

text: (F

text: n)

text: as

text: shown

text: by

text: Zagier

text: and

text: their

text: generating

text: function

text: n

text: x)k),

text: n

text: n

text: derived

text: by

text: Zagier

text: is

text: not

text: even

text: D-ﬁnite

text: Computer

text: Science

text: Institute,

text: Faculty

text: of

text: Mathematics

text: and

text: Physics,

text: Charles

text: University,

text: Malostransk´e

text: n´amˇest´ı

text: Prague,

text: jelinek@iuuk.mff.cuni.cz.

text: Supported

text: by

text: project

text: CE-ITI

text: of

text: the

text: Czech

text: Science

text: Foundation.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 1 1 1 1 2 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 1 1 2 2 1 2 1 1 2 2 Figure 8: Illustration of the symmetric joint distribution of the number of wNE-cells and the weight of last column over Fishburn matrices. Left: the Fishburn matrices of weight 5, with 3 wNE-cells, and last column of weight 2. Right: the Fishburn matrices of weight 5, with 2 wNE-cells, and last column of weight 3. The wNE-cells are shaded. 3.1 Proofs To prove Theorems 3.6 and 3.7, it is more convenient to work with Fishburn matrices rather than relational structures, and to interpret the relevant statistics using Observation 3.5. Recall that an interval order is primitive if it has no two indistinguishable elements. Primitive interval orders correspond to Fishburn matrices whose entries are equal to 0 or 1; we call such matrices primitive Fishburn matrices. An inﬂation of a primitive Fishburn matrix M is an operation which replaces the value of each 1-cell of M (i.e., a cell of value 1) by a positive integer, while the 0-cells are left unchanged. Clearly, by inﬂating a primitive Fishburn matrix we again obtain a Fishburn matrix, and any Fishburn matrix can be uniquely obtained by inﬂating a primitive Fishburn matrix. Another useful operation on primitive Fishburn matrices is the extension. Informally speaking, it creates (cid:48) a primitive Fishburn matrix P with k + 1 columns from a primitive Fishburn matrix P with k-columns, by splitting the last column of P into two new columns. Formally, suppose that P = (P )k is a k-by-k (cid:48) (cid:48) i,j i,j=1 primitive Fishburn matrix. We say that a (k + 1)-by-(k + 1) matrix P = (P )k+1 is an extension of P, if (cid:48) i,j i,j=1 P has the following properties (see Figure 9): • (cid:48) ≤ (cid:48) The last row of P consists of k 0-cells followed by a 1-cell. In other words, for j k we have P = 0, k+1,j while P = 1. k+1,k+1 • ≤ (cid:48) − (cid:48) For every j < k and for every i j, we have P = P . That is, the ﬁrst k 1 columns of P are − i,j i,j identical to the ﬁrst k 1 columns of P, except for an extra 0-cell in the last row. • (cid:48) (cid:48) If P = 0 for some i, then P = P = 0. That is, each 0-cell in the last column of P gives rise i,k i,k i,k+1 (cid:48) to two 0-cells in the same row and in the last two columns of P . • (cid:48) (cid:48) If P = 1, then there are three options for the values of P and P : i,k i,k i,k+1 (cid:48) (cid:48) (cid:48) (cid:48) 1. P = P = 1. In such case we say that the 1-cell P is duplicated into P and P . i,k i,k+1 i,k i,k i,k+1 (cid:48) (cid:48) (cid:48) 2. P = 0 and P = 1. We then say that P is shifted into P . i,k i,k+1 i,k i,k+1 (cid:48) (cid:48) 3. P = 1 and P = 0. We then say that P is ignored by the extension. i,k i,k+1 i,k (cid:48) We say that an extension of P into P is valid, if there is at least one 1-cell in the penultimate column of (cid:48) P , or equivalently, if at least one 1-cell in the last column of P has been duplicated or ignored. It is easy 15 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Figure

text: Illustration

label: caption

text: of

text: the

text: symmetric

text: joint

text: distribution

text: of

text: the

text: number

text: of

text: wNE-cells

text: and

text: the

text: weight

text: of

text: last

text: column

text: over

text: Fishburn

text: matrices.

text: Left:

text: the

text: Fishburn

text: matrices

text: of

text: weight

text: with

text: wNE-cells,

text: and

text: last

text: column

text: of

text: weight

text: Right:

text: the

text: Fishburn

text: matrices

text: of

text: weight

text: with

text: wNE-cells,

text: and

text: last

text: column

text: of

text: weight

text: The

text: wNE-cells

text: are

text: shaded.

text: Proofs

text: To

text: prove

text: Theorems

text: and

text: it

text: is

text: more

text: convenient

text: to

text: work

text: with

text: Fishburn

text: matrices

text: rather

text: than

text: relational

text: structures,

text: and

text: to

text: interpret

text: the

text: relevant

text: statistics

text: using

text: Observation

text: Recall

text: that

text: an

text: interval

text: order

text: is

text: primitive

text: if

text: it

text: has

text: no

text: two

text: indistinguishable

text: elements.

text: Primitive

text: interval

text: orders

text: correspond

text: to

text: Fishburn

text: matrices

text: whose

text: entries

text: are

text: equal

text: to

text: or

text: we

text: call

text: such

text: matrices

text: primitive

text: Fishburn

text: matrices.

text: An

text: inﬂation

text: of

text: a

text: primitive

text: Fishburn

text: matrix

text: M

text: is

text: an

text: operation

text: which

text: replaces

text: the

text: value

text: of

text: each

text: of

text: M

text: (i.e.,

text: a

text: cell

text: of

text: value

text: by

text: a

text: positive

text: integer,

text: while

text: the

text: are

text: left

text: unchanged.

text: Clearly,

text: by

text: inﬂating

text: a

text: primitive

text: Fishburn

text: matrix

text: we

text: again

text: obtain

text: a

text: Fishburn

text: matrix,

text: and

text: any

text: Fishburn

text: matrix

text: can

text: be

text: uniquely

text: obtained

text: by

text: inﬂating

text: a

text: primitive

text: Fishburn

text: matrix.

text: Another

text: useful

text: operation

text: on

text: primitive

text: Fishburn

text: matrices

text: is

text: the

text: extension.

text: Informally

text: speaking,

text: it

text: creates

text: a

text: primitive

text: Fishburn

text: matrix

text: P

text: with

text: k

text: columns

text: from

text: a

text: primitive

text: Fishburn

text: matrix

text: P

text: with

text: k-columns,

text: by

text: splitting

text: the

text: last

text: column

text: of

text: P

text: into

text: two

text: new

text: columns.

text: Formally,

text: suppose

text: that

text: P

text: (P

text: )k

text: is

text: a

text: k-by-k

text: i,j

text: primitive

text: Fishburn

text: matrix.

text: We

text: say

text: that

text: a

text: (k

text: matrix

text: P

text: (P

text: is

text: an

text: extension

text: of

text: P,

text: if

text: i,j

text: P

text: has

text: the

text: following

text: properties

text: (see

text: Figure

text: The

text: last

text: row

text: of

text: P

text: consists

text: of

text: k

text: followed

text: by

text: a

text: In

text: other

text: words,

text: for

text: j

text: k

text: we

text: have

text: P

text: while

text: P

text: For

text: every

text: j

text: k

text: and

text: for

text: every

text: i

text: j,

text: we

text: have

text: P

text: P

text: That

text: is,

text: the

text: ﬁrst

text: k

text: columns

text: of

text: P

text: are

text: i,j

text: i,j

text: identical

text: to

text: the

text: ﬁrst

text: k

text: columns

text: of

text: P,

text: except

text: for

text: an

text: extra

text: in

text: the

text: last

text: row.

text: If

text: P

text: for

text: some

text: i,

text: then

text: P

text: P

text: That

text: is,

text: each

text: in

text: the

text: last

text: column

text: of

text: P

text: gives

text: rise

text: i,k

text: i,k

text: to

text: two

text: in

text: the

text: same

text: row

text: and

text: in

text: the

text: last

text: two

text: columns

text: of

text: P

text: If

text: P

text: then

text: there

text: are

text: three

text: options

text: for

text: the

text: values

text: of

text: P

text: and

text: P

text: i,k

text: i,k

text: P

text: P

text: In

text: such

text: case

text: we

text: say

text: that

text: the

text: P

text: is

text: duplicated

text: into

text: P

text: and

text: P

text: i,k

text: i,k

text: i,k

text: P

text: and

text: P

text: We

text: then

text: say

text: that

text: P

text: is

text: shifted

text: into

text: P

text: i,k

text: i,k

text: P

text: and

text: P

text: We

text: then

text: say

text: that

text: P

text: is

text: ignored

text: by

text: the

text: extension.

text: i,k

text: i,k

text: We

text: say

text: that

text: an

text: extension

text: of

text: P

text: into

text: P

text: is

text: valid,

text: if

text: there

text: is

text: at

text: least

text: one

text: in

text: the

text: penultimate

text: column

text: of

text: P

text: or

text: equivalently,

text: if

text: at

text: least

text: one

text: in

text: the

text: last

text: column

text: of

text: P

text: has

text: been

text: duplicated

text: or

text: ignored.

text: It

text: is

text: easy

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 1 1 1 1 1 1 1 1 1 1 DIS 1 1 1 1 Figure 9: Example of an extension of a primitive Fishburn matrix. The word DIS is the code of the extension. (cid:48) (cid:48) to see that if P is a valid extension of a primitive Fishburn matrix P, then P is itself a primitive Fishburn (cid:48) matrix, and conversely, any primitive Fishburn matrix P with at least two columns is a valid extension of a unique primitive Fishburn matrix P. Note that a primitive Fishburn matrix P whose last column has weight m has exactly 3m extensions; − (cid:48) one of them is invalid and 3m 1 are valid. For convenience, we will represent each extension P of P by a · · · { } word w = w w of length m over the alphabet D, S, I , deﬁned as follows: suppose that c , c , . . . , c 1 m 1 2 m are the 1-cells in the last column of P, listed in top-to-bottom order. Then w is equal to D (or S or I), i (cid:48) if the cell c is duplicated (or shifted, or ignored, respectively) in the extension P . We will call w the code i (cid:48) (cid:48) of the extension from P to P . Notice that the 1-cell in the bottom-right corner of P is not represented by any symbol of w. · · · · · · Given a word w = w w w of length m, the reverse of w, denoted by w, is the word w w − w . 1 2 m m m 1 1 Observation 3.8. Let P be a k-by-k primitive Fishburn matrix with m 1-cells in the last column, and let (cid:48) · · · (cid:48) P be its valid extension, with code w = w w . Let c be a 1-cell in the j-th column of P . 1 m • Suppose that j < k, which implies, in particular, that c is also a 1-cell in P. Then c is an sNE-cell of (cid:48) (cid:48) (cid:48) (cid:48) P (or wNE-cell of P , or sSE-cell of P , or wNE-cell of P ) if and only if it is a sNE-cell of P (or wNE-cell of P, or sSE-cell of P, or wNE-cell of P, respectively). • Suppose that j = k, which means that c is also a 1-cell in P, and this 1-cell was duplicated or ignored (cid:48) by the extension from P to P . Suppose c is the i-th 1-cell in the last column of P, counted from the − − top (i.e., there are i 1 1-cells above c and m i 1-cells below c in the last column of P). Then c is (cid:48) (cid:48) a wNE-cell of P if and only if i = 1 and w = I, while c is a wSE-cell of P if and only if i = m and 1 (cid:48) w = I. Furthermore, c is an sNE-cell of P if and only if all the 1-cells of P above it were ignored m · · · (cid:48) (i.e., w = w = = w − = I), while c is an sSE-cell of P if and only if all the 1-cells of P below 1 2 i 1 · · · it were ignored (i.e., w = w = = w = I). i+1 i+2 m • (cid:48) If j = k + 1, i.e., c is in the last column of P , then c is an sNE-cell and also an sSE-cell. Moreover, (cid:48) c is a wNE-cell if and only if it is the topmost 1-cell of the last column of P , and it is a wSE-cell if (cid:48) and only if it is the bottommost 1-cell of the last column of P . 3.1.1 Proof of Theorem 3.6 To prove Theorem 3.6, we ﬁrst describe the involution φ, and then verify that it has the required properties. Let M be a k-by-k Fishburn matrix. As explained above, M can be constructed in a unique way from − the 1-by-1 matrix 1 by a sequence of k 1 valid extensions followed by an inﬂation. In particular, there is a sequence of matrices P , P , . . . , P , M, where P = 1 , for i > 1 the matrix P is an extension of P − , 1 2 k 1 i i 1 and M is an inﬂation of P . k Deﬁne a new sequence P , . . . , P as follows: • 1 k P = P = 1 . 1 1 16 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DIS

text: Figure

text: Example

text: of

text: an

text: extension

text: of

text: a

text: primitive

text: Fishburn

text: matrix.

text: The

text: word

text: DIS

text: is

text: the

text: code

text: of

text: the

text: extension.

text: to

text: see

text: that

text: if

text: P

text: is

text: a

text: valid

text: extension

text: of

text: a

text: primitive

text: Fishburn

text: matrix

text: P,

text: then

text: P

text: is

text: itself

text: a

text: primitive

text: Fishburn

text: matrix,

text: and

text: conversely,

text: any

text: primitive

text: Fishburn

text: matrix

text: P

text: with

text: at

text: least

text: two

text: columns

text: is

text: a

text: valid

text: extension

text: of

text: a

text: unique

text: primitive

text: Fishburn

text: matrix

text: P.

text: Note

text: that

text: a

text: primitive

text: Fishburn

text: matrix

text: P

text: whose

text: last

text: column

text: has

text: weight

text: m

text: has

text: exactly

text: extensions;

text: one

text: of

text: them

text: is

text: invalid

text: and

text: are

text: valid.

text: For

text: convenience,

text: we

text: will

text: represent

text: each

text: extension

text: P

text: of

text: P

text: by

text: a

text: word

text: w

text: w

text: w

text: of

text: length

text: m

text: over

text: the

text: alphabet

text: D,

text: S,

text: I

text: deﬁned

text: as

text: follows:

text: suppose

text: that

text: c

text: c

text: c

text: m

text: m

text: are

text: the

text: in

text: the

text: last

text: column

text: of

text: P,

text: listed

text: in

text: top-to-bottom

text: order.

text: Then

text: w

text: is

text: equal

text: to

text: D

text: (or

text: S

text: or

text: I),

text: i

text: if

text: the

text: cell

text: c

text: is

text: duplicated

text: (or

text: shifted,

text: or

text: ignored,

text: respectively)

text: in

text: the

text: extension

text: P

text: We

text: will

text: call

text: w

text: the

text: code

text: i

text: of

text: the

text: extension

text: from

text: P

text: to

text: P

text: Notice

text: that

text: the

text: in

text: the

text: bottom-right

text: corner

text: of

text: P

text: is

text: not

text: represented

text: by

text: any

text: symbol

text: of

text: w.

text: Given

text: a

text: word

text: w

text: w

text: w

text: w

text: of

text: length

text: m,

text: the

text: reverse

text: of

text: w,

text: denoted

text: by

text: w,

text: is

text: the

text: word

text: w

text: w

text: w

text: m

text: m

text: m

text: Observation

text: Let

text: P

text: be

text: a

text: k-by-k

text: primitive

text: Fishburn

text: matrix

text: with

text: m

text: in

text: the

text: last

text: column,

text: and

text: let

text: P

text: be

text: its

text: valid

text: extension,

text: with

text: code

text: w

text: w

text: w

text: Let

text: c

text: be

text: a

text: in

text: the

text: j-th

text: column

text: of

text: P

text: m

text: Suppose

text: that

text: j

text: k,

text: which

text: implies,

text: in

text: particular,

text: that

text: c

text: is

text: also

text: a

text: in

text: P.

text: Then

text: c

text: is

text: an

text: sNE-cell

text: of

text: P

text: (or

text: wNE-cell

text: of

text: P

text: or

text: sSE-cell

text: of

text: P

text: or

text: wNE-cell

text: of

text: P

text: if

text: and

text: only

text: if

text: it

text: is

text: a

text: sNE-cell

text: of

text: P

text: (or

text: wNE-cell

text: of

text: P,

text: or

text: sSE-cell

text: of

text: P,

text: or

text: wNE-cell

text: of

text: P,

text: respectively).

text: Suppose

text: that

text: j

text: k,

text: which

text: means

text: that

text: c

text: is

text: also

text: a

text: in

text: P,

text: and

text: this

text: was

text: duplicated

text: or

text: ignored

text: by

text: the

text: extension

text: from

text: P

text: to

text: P

text: Suppose

text: c

text: is

text: the

text: i-th

text: in

text: the

text: last

text: column

text: of

text: P,

text: counted

text: from

text: the

text: top

text: (i.e.,

text: there

text: are

text: i

text: above

text: c

text: and

text: m

text: i

text: below

text: c

text: in

text: the

text: last

text: column

text: of

text: P).

text: Then

text: c

text: is

text: a

text: wNE-cell

text: of

text: P

text: if

text: and

text: only

text: if

text: i

text: and

text: w

text: I,

text: while

text: c

text: is

text: a

text: wSE-cell

text: of

text: P

text: if

text: and

text: only

text: if

text: i

text: m

text: and

text: w

text: I.

text: Furthermore,

text: c

text: is

text: an

text: sNE-cell

text: of

text: P

text: if

text: and

text: only

text: if

text: all

text: the

text: of

text: P

text: above

text: it

text: were

text: ignored

text: m

text: (i.e.,

text: w

text: w

text: w

text: I),

text: while

text: c

text: is

text: an

text: sSE-cell

text: of

text: P

text: if

text: and

text: only

text: if

text: all

text: the

text: of

text: P

text: below

text: i

text: it

text: were

text: ignored

text: (i.e.,

text: w

text: w

text: w

text: I).

text: m

text: If

text: j

text: k

text: i.e.,

text: c

text: is

text: in

text: the

text: last

text: column

text: of

text: P

text: then

text: c

text: is

text: an

text: sNE-cell

text: and

text: also

text: an

text: sSE-cell.

text: Moreover,

text: c

text: is

text: a

text: wNE-cell

text: if

text: and

text: only

text: if

text: it

text: is

text: the

text: topmost

text: of

text: the

text: last

text: column

text: of

text: P

text: and

text: it

text: is

text: a

text: wSE-cell

text: if

text: and

text: only

text: if

text: it

text: is

text: the

text: bottommost

text: of

text: the

text: last

text: column

text: of

text: P

text: Proof

text: of

text: Theorem

text: To

text: prove

text: Theorem

text: we

text: ﬁrst

text: describe

text: the

text: involution

text: φ,

text: and

text: then

text: verify

text: that

text: it

text: has

text: the

text: required

text: properties.

text: Let

text: M

text: be

text: a

text: k-by-k

text: Fishburn

text: matrix.

text: As

text: explained

text: above,

text: M

text: can

text: be

text: constructed

text: in

text: a

text: unique

text: way

text: from

text: the

text: matrix

text: by

text: a

text: sequence

text: of

text: k

text: valid

text: extensions

text: followed

text: by

text: an

text: inﬂation.

text: In

text: particular,

text: there

text: is

text: a

text: sequence

text: of

text: matrices

text: P

text: P

text: P

text: M,

text: where

text: P

text: for

text: i

text: the

text: matrix

text: P

text: is

text: an

text: extension

text: of

text: P

text: k

text: i

text: i

text: and

text: M

text: is

text: an

text: inﬂation

text: of

text: P

text: k

text: Deﬁne

text: a

text: new

text: sequence

text: P

text: P

text: as

text: follows:

text: k

text: P

text: P

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 + 2 3 + 1 N Figure 1: The posets 2+2 (left), 3+1 (center), and N (right), which will frequently play the role of forbidden patterns in this paper. ∗ Fishburn numbers. The N-free posets are also known as series-parallel posets. They are exactly the posets that can be constructed from a single-element poset by a repeated application of direct sums and disjoint unions. The (3+1)-free posets do not seem to admit such a simple characterization, and their structural description is the topic of ongoing research [20, 27, 33, 34]. The (2+2, 3+1)-free posets, i.e., the posets avoiding both 2+2 and 3+1, are also known as semiorders, and have been introduced by Luce [28] in the 1950s. They are enumerated by Catalan numbers [24]. The (2+2, N)-free posets are enumerated by Catalan numbers as well, as shown by Disanto et al. [7, 10]. 1.2 Interval orders and Fishburn matrices Let us brieﬂy summarize several known facts about interval orders and their representations. A detailed treatment of this topic appears, e.g., in Fishburn’s book [18]. ≺ Let R = (X, ) be a poset. An interval representation of R is a mapping I that associates to every ∈ ∈ element x X a closed interval I(x) = [l , r ] in such a way that for any two elements x, y X, we have ≺ x x x y if and only if r < l . Note that we allow the intervals I(x) to be degenerate, i.e., to consist of a x y single point. A poset has an interval representation if and only if it is (2+2)-free, i.e., if it is an interval order. An interval representation I is minimal, if it satisﬁes these conditions: • ∈ for every x X, the endpoints of I(x) are positive integers • ∈ N ∈ { } there is a positive integer m such that for every k 1, 2, . . . , m there is an interval I(x) whose right endpoint is k, as well as an interval I(y) whose left endpoint is k. Each interval order R has a unique minimal interval representation. The integer m, which corresponds to the number of distinct endpoints in the minimal representation, is known as the magnitude of R. Note ∈ that x X is a minimal element of R if and only if the left endpoint of I(x) is equal to 1, and x is a maximal element if and only if the right endpoint of I(x) is equal to m. ≺ ∈ Two elements x, y of a poset R = (X, ) are indistinguishable if for every element z X we have the ≺ ⇐⇒ ≺ ≺ ⇐⇒ ≺ equivalences x z y z and z x z y. In an interval order R with a minimal interval representation I, two elements x, y are indistinguishable if and only if I(x) = I(y). An interval order is primitive if it has no two indistinguishable elements. Every interval order R can be uniquely constructed (cid:48) ∈ (cid:48) from a primitive interval order R by replacing each element x R by a group of indistinguishable elements containing x. Given a matrix M, we use the term cell (i, j) of M to refer to the entry in the i-th row and j-th column of M, and we let M denote its value. We assume that the rows of a matrix are numbered top to bottom, i,j † that is, the top row has number 1. The weight of a matrix M is the sum of its entries . Similarly, the weight of a row (or a column, or a diagonal) of a matrix is the sum of the entries in this row (or column or diagonal). A Fishburn matrix is an upper-triangular square matrix of nonnegative integers whose every row and × every column has nonzero weight. In other words, an m m matrix M of nonnegative integers is a Fishburn matrix if it satisﬁes these conditions: ∗ Beware that some authors (e.g. Khamis [23]) give the term ‘N-free poset’ a meaning subtly diﬀerent from ours. † Some earlier papers use the term size of M instead of weight of M. However, in other contexts the term ‘size’ often refers to the number of rows of a matrix. We therefore prefer to use the less ambiguous term ‘weight’ in this paper. 3 ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: N

text: Figure

text: The

text: posets

text: (left),

text: (center),

text: and

text: N

text: (right),

text: which

text: will

text: frequently

text: play

text: the

text: role

text: of

text: forbidden

text: patterns

text: in

text: this

text: paper.

text: Fishburn

text: numbers.

text: The

text: N-free

text: posets

text: are

text: also

text: known

text: as

text: series-parallel

text: posets.

text: They

text: are

text: exactly

text: the

text: posets

text: that

text: can

text: be

text: constructed

text: from

text: a

text: single-element

text: poset

text: by

text: a

text: repeated

text: application

text: of

text: direct

text: sums

text: and

text: disjoint

text: unions.

text: The

text: posets

text: do

text: not

text: seem

text: to

text: admit

text: such

text: a

text: simple

text: characterization,

text: and

text: their

text: structural

text: description

text: is

text: the

text: topic

text: of

text: ongoing

text: research

text: The

text: posets,

text: i.e.,

text: the

text: posets

text: avoiding

text: both

text: and

text: are

text: also

text: known

text: as

text: semiorders,

text: and

text: have

text: been

text: introduced

text: by

text: Luce

text: in

text: the

text: They

text: are

text: enumerated

text: by

text: Catalan

text: numbers

text: The

text: N)-free

text: posets

text: are

text: enumerated

text: by

text: Catalan

text: numbers

text: as

text: well,

text: as

text: shown

text: by

text: Disanto

text: et

text: al.

text: Interval

text: orders

text: and

text: Fishburn

text: matrices

text: Let

text: us

text: brieﬂy

text: summarize

text: several

text: known

text: facts

text: about

text: interval

text: orders

text: and

text: their

text: representations.

text: A

text: detailed

text: treatment

text: of

text: this

text: topic

text: appears,

text: e.g.,

text: in

text: Fishburn’s

text: book

text: Let

text: R

text: (X,

text: be

text: a

text: poset.

text: An

text: interval

text: representation

text: of

text: R

text: is

text: a

text: mapping

text: I

text: that

text: associates

text: to

text: every

text: element

text: x

text: X

text: a

text: closed

text: interval

text: I(x)

text: [l

text: r

text: in

text: such

text: a

text: way

text: that

text: for

text: any

text: two

text: elements

text: x,

text: y

text: X,

text: we

text: have

text: x

text: x

text: x

text: y

text: if

text: and

text: only

text: if

text: r

text: l

text: Note

text: that

text: we

text: allow

text: the

text: intervals

text: I(x)

text: to

text: be

text: degenerate,

text: i.e.,

text: to

text: consist

text: of

text: a

text: x

text: y

text: single

text: point.

text: A

text: poset

text: has

text: an

text: interval

text: representation

text: if

text: and

text: only

text: if

text: it

text: is

text: i.e.,

text: if

text: it

text: is

text: an

text: interval

text: order.

text: An

text: interval

text: representation

text: I

text: is

text: minimal,

text: if

text: it

text: satisﬁes

text: these

text: conditions:

text: for

text: every

text: x

text: X,

text: the

text: endpoints

text: of

text: I(x)

text: are

text: positive

text: integers

text: N

text: there

text: is

text: a

text: positive

text: integer

text: m

text: such

text: that

text: for

text: every

text: k

text: m

text: there

text: is

text: an

text: interval

text: I(x)

text: whose

text: right

text: endpoint

text: is

text: k,

text: as

text: well

text: as

text: an

text: interval

text: I(y)

text: whose

text: left

text: endpoint

text: is

text: k.

text: Each

text: interval

text: order

text: R

text: has

text: a

text: unique

text: minimal

text: interval

text: representation.

text: The

text: integer

text: m,

text: which

text: corresponds

text: to

text: the

text: number

text: of

text: distinct

text: endpoints

text: in

text: the

text: minimal

text: representation,

text: is

text: known

text: as

text: the

text: magnitude

text: of

text: R.

text: Note

text: that

text: x

text: X

text: is

text: a

text: minimal

text: element

text: of

text: R

text: if

text: and

text: only

text: if

text: the

text: left

text: endpoint

text: of

text: I(x)

text: is

text: equal

text: to

text: and

text: x

text: is

text: a

text: maximal

text: element

text: if

text: and

text: only

text: if

text: the

text: right

text: endpoint

text: of

text: I(x)

text: is

text: equal

text: to

text: m.

text: Two

text: elements

text: x,

text: y

text: of

text: a

text: poset

text: R

text: (X,

text: are

text: indistinguishable

text: if

text: for

text: every

text: element

text: z

text: X

text: we

text: have

text: the

text: equivalences

text: x

text: z

text: y

text: z

text: and

text: z

text: x

text: z

text: y.

text: In

text: an

text: interval

text: order

text: R

text: with

text: a

text: minimal

text: interval

text: representation

text: I,

text: two

text: elements

text: x,

text: y

text: are

text: indistinguishable

text: if

text: and

text: only

text: if

text: I(x)

text: I(y).

text: An

text: interval

text: order

text: is

text: primitive

text: if

text: it

text: has

text: no

text: two

text: indistinguishable

text: elements.

text: Every

text: interval

text: order

text: R

text: can

text: be

text: uniquely

text: constructed

text: from

text: a

text: primitive

text: interval

text: order

text: R

text: by

text: replacing

text: each

text: element

text: x

text: R

text: by

text: a

text: group

text: of

text: indistinguishable

text: elements

text: containing

text: x.

text: Given

text: a

text: matrix

text: M,

text: we

text: use

text: the

text: term

text: cell

text: (i,

text: j)

text: of

text: M

text: to

text: refer

text: to

text: the

text: entry

text: in

text: the

text: i-th

text: row

text: and

text: j-th

text: column

text: of

text: M,

text: and

text: we

text: let

text: M

text: denote

text: its

text: value.

text: We

text: assume

text: that

text: the

text: rows

text: of

text: a

text: matrix

text: are

text: numbered

text: top

text: to

text: bottom,

text: i,j

text: that

text: is,

text: the

text: top

text: row

text: has

text: number

text: The

text: weight

text: of

text: a

text: matrix

text: M

text: is

text: the

text: sum

text: of

text: its

text: entries

text: Similarly,

text: the

text: weight

text: of

text: a

text: row

text: (or

text: a

text: column,

text: or

text: a

text: diagonal)

text: of

text: a

text: matrix

text: is

text: the

text: sum

text: of

text: the

text: entries

text: in

text: this

text: row

text: (or

text: column

text: or

text: diagonal).

text: A

text: Fishburn

text: matrix

text: is

text: an

text: upper-triangular

text: square

text: matrix

text: of

text: nonnegative

text: integers

text: whose

text: every

text: row

text: and

text: every

text: column

text: has

text: nonzero

text: weight.

text: In

text: other

text: words,

text: an

text: m

text: m

text: matrix

text: M

text: of

text: nonnegative

text: integers

text: is

text: a

text: Fishburn

text: matrix

text: if

text: it

text: satisﬁes

text: these

text: conditions:

text: Beware

text: that

text: some

text: authors

text: (e.g.

text: Khamis

text: give

text: the

text: term

text: ‘N-free

text: poset’

text: a

text: meaning

text: subtly

text: diﬀerent

text: from

text: ours.

text: Some

text: earlier

text: papers

text: use

text: the

text: term

text: size

text: of

text: M

text: instead

text: of

text: weight

text: of

text: M.

text: However,

text: in

text: other

text: contexts

text: the

text: term

text: ‘size’

text: often

text: refers

text: to

text: the

text: number

text: of

text: rows

text: of

text: a

text: matrix.

text: We

text: therefore

text: prefer

text: to

text: use

text: the

text: less

text: ambiguous

text: term

text: ‘weight’

text: in

text: this

text: paper.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [13] M. Dukes, S. Kitaev, J. Remmel, and E. Steingr´ımsson. Enumerating (2 + 2)-free posets by indistinguishable elements. Journal of Combinatorics, 2(1):139–163, 2011. [14] M. Dukes and R. Parviainen. Ascent sequences and upper triangular matrices containing non-negative integers. Electronic J. Combin., 17(R53), 2010. [15] P. Duncan and E. Steingr´ımsson. Pattern avoidance in ascent sequences. Electronic J. Combin., 18(P226), 2011. [16] P. C. Fishburn. Intransitive indiﬀerence with unequal indiﬀerence intervals. Journal of Mathematical Psychology, 7(1):144–149, 1970. [17] P. C. Fishburn. Interval graphs and interval orders. Discrete Math., 55(2):135–149, 1985. [18] P. C. Fishburn. Interval orders and interval graphs: A study of partially ordered sets. John Wiley & Sons, 1985. [19] H. Friedman and D. Tamari. Probl`emes d’associativit´e: Une structure de treillis ﬁnis induite par une loi demi-associative. J. Comb. Theory, 2(3):215–242, 1967. [20] M. Guay-Paquet, A. H. Morales, and E. Rowland. Structure and enumeration of (3+1)-free posets. Ann. Comb., 18(4):645–674, 2014. [21] S. Huang and D. Tamari. Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law. J. Comb. Theory A, 13(1):7–13, 1972. [22] V. Jel´ınek. Counting general and self-dual interval orders. J. Comb. Theory A, 119(3):599–614, 2012. [23] S. M. Khamis. Height counting of unlabeled interval and N-free posets. Discrete Math., 275(1-3):165–175, 2004. [24] K.H. Kim and F.W. Roush. Enumeration of isomorphism classes of semiorders. Journal of Combinatorics, Information & System Sciences, 3:58–61, 1978. [25] S. Kitaev and J. B. Remmel. Enumerating (2+2)-free posets by the number of minimal elements and other statistics. Disc. Appl. Math., 159(17):2098–2108, 2011. [26] P. Levande. Fishburn diagrams, Fishburn numbers and their reﬁned generating functions. J. Comb. Theory A, 120(1):194–217, 2013. [27] J. B. Lewis and Y. X. Zhang. Enumeration of graded (3+1)-avoiding posets. J. Comb. Theory A, 120(6):1305– 1327, 2013. [28] R. D. Luce. Semiorders and a theory of utility discrimination. Econometrica, 24(2):pp. 178–191, 1956. [29] T. Mansour and M. Shattuck. Some enumerative results related to ascent sequences. Discrete Math., 315- 316:29–41, 2014. [30] OEIS Foundation, Inc. The On-Line Encyclopedia of Integer Sequences. http://oeis.org/, 2011. [31] R. Parviainen. Wilf classiﬁcation of bi-vincular permutation patterns. arXiv:0910.5103, 2009. [32] M. Rubey. Increasing and decreasing sequences in ﬁllings of moon polyominoes. Adv. Appl. Math., 47:57–87, 2011. [33] M. Skandera. A Characterization of (3+1)-Free Posets. J. Comb. Theory A, 93(2):231–241, 2001. [34] M. Skandera and B. Reed. Total nonnegativity and (3 + 1)-free posets. J. Comb. Theory A, 103(2):237–256, 2003. ∼ [35] R. P. Stanley. Catalan Addendum (version of May 2013). www-math.mit.edu/ rstan/ec/catadd.pdf. [36] A. Stoimenow. Enumeration of chord diagrams and an upper bound for Vassiliev invariants. J. Knot Theory Ramiﬁcations, 7:93–114, 1998. [37] S. H. F. Yan. On a conjecture about enumerating (2+2)-free posets. Eur. J. Combin., 32(2):282–287, 2011. [38] D. Zagier. Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology, 40(5):945– 960, 2001. 22

label: reference

text: M.

text: Dukes,

text: S.

text: Kitaev,

text: J.

text: Remmel,

text: and

text: E.

text: Steingr´ımsson.

text: Enumerating

text: posets

text: by

text: indistinguishable

text: elements.

text: Journal

text: of

text: Combinatorics,

text: M.

text: Dukes

text: and

text: R.

text: Parviainen.

text: Ascent

text: sequences

text: and

text: upper

text: triangular

text: matrices

text: containing

text: non-negative

text: integers.

text: Electronic

text: J.

text: Combin.,

text: P.

text: Duncan

text: and

text: E.

text: Steingr´ımsson.

text: Pattern

text: avoidance

text: in

text: ascent

text: sequences.

text: Electronic

text: J.

text: Combin.,

text: P.

text: C.

text: Fishburn.

text: Intransitive

text: indiﬀerence

text: with

text: unequal

text: indiﬀerence

text: intervals.

text: Journal

text: of

text: Mathematical

text: Psychology,

text: P.

text: C.

text: Fishburn.

text: Interval

text: graphs

text: and

text: interval

text: orders.

text: Discrete

text: Math.,

text: P.

text: C.

text: Fishburn.

text: Interval

text: orders

text: and

text: interval

text: graphs:

text: A

text: study

text: of

text: partially

text: ordered

text: sets.

text: John

text: Wiley

text: Sons,

text: H.

text: Friedman

text: and

text: D.

text: Tamari.

text: Probl`emes

text: d’associativit´e:

text: Une

text: structure

text: de

text: treillis

text: ﬁnis

text: induite

text: par

text: une

text: loi

text: demi-associative.

text: J.

text: Comb.

text: Theory,

text: M.

text: Guay-Paquet,

text: A.

text: H.

text: Morales,

text: and

text: E.

text: Rowland.

text: Structure

text: and

text: enumeration

text: of

text: posets.

text: Ann.

text: Comb.,

text: S.

text: Huang

text: and

text: D.

text: Tamari.

text: Problems

text: of

text: associativity:

text: A

text: simple

text: proof

text: for

text: the

text: lattice

text: property

text: of

text: systems

text: ordered

text: by

text: a

text: semi-associative

text: law.

text: J.

text: Comb.

text: Theory

text: A,

text: V.

text: Jel´ınek.

text: Counting

text: general

text: and

text: self-dual

text: interval

text: orders.

text: J.

text: Comb.

text: Theory

text: A,

text: S.

text: M.

text: Khamis.

text: Height

text: counting

text: of

text: unlabeled

text: interval

text: and

text: N-free

text: posets.

text: Discrete

text: Math.,

text: K.H.

text: Kim

text: and

text: F.W.

text: Roush.

text: Enumeration

text: of

text: isomorphism

text: classes

text: of

text: semiorders.

text: Journal

text: of

text: Combinatorics,

text: Information

text: System

text: Sciences,

text: S.

text: Kitaev

text: and

text: J.

text: B.

text: Remmel.

text: Enumerating

text: posets

text: by

text: the

text: number

text: of

text: minimal

text: elements

text: and

text: other

text: statistics.

text: Disc.

text: Appl.

text: Math.,

text: P.

text: Levande.

text: Fishburn

text: diagrams,

text: Fishburn

text: numbers

text: and

text: their

text: reﬁned

text: generating

text: functions.

text: J.

text: Comb.

text: Theory

text: A,

text: J.

text: B.

text: Lewis

text: and

text: Y.

text: X.

text: Zhang.

text: Enumeration

text: of

text: graded

text: posets.

text: J.

text: Comb.

text: Theory

text: A,

text: R.

text: D.

text: Luce.

text: Semiorders

text: and

text: a

text: theory

text: of

text: utility

text: discrimination.

text: Econometrica,

text: T.

text: Mansour

text: and

text: M.

text: Shattuck.

text: Some

text: enumerative

text: results

text: related

text: to

text: ascent

text: sequences.

text: Discrete

text: Math.,

text: OEIS

text: Foundation,

text: Inc.

text: The

text: On-Line

text: Encyclopedia

text: of

text: Integer

text: Sequences.

text: R.

text: Parviainen.

text: Wilf

text: classiﬁcation

text: of

text: bi-vincular

text: permutation

text: patterns.

text: M.

text: Rubey.

text: Increasing

text: and

text: decreasing

text: sequences

text: in

text: ﬁllings

text: of

text: moon

text: polyominoes.

text: Adv.

text: Appl.

text: Math.,

text: M.

text: Skandera.

text: A

text: Characterization

text: of

text: Posets.

text: J.

text: Comb.

text: Theory

text: A,

text: M.

text: Skandera

text: and

text: B.

text: Reed.

text: Total

text: nonnegativity

text: and

text: posets.

text: J.

text: Comb.

text: Theory

text: A,

text: R.

text: P.

text: Stanley.

text: Catalan

text: Addendum

text: (version

text: of

text: May

text: www-math.mit.edu/

text: A.

text: Stoimenow.

text: Enumeration

text: of

text: chord

text: diagrams

text: and

text: an

text: upper

text: bound

text: for

text: Vassiliev

text: invariants.

text: J.

text: Knot

text: Theory

text: Ramiﬁcations,

text: S.

text: H.

text: F.

text: Yan.

text: On

text: a

text: conjecture

text: about

text: enumerating

text: posets.

text: Eur.

text: J.

text: Combin.,

text: D.

text: Zagier.

text: Vassiliev

text: invariants

text: and

text: a

text: strange

text: identity

text: related

text: to

text: the

text: Dedekind

text: eta-function.

text: Topology,


text: f g e 1 1 a c 1 c d e f 1 2 b d g 1 a b 1 2 3 4 Figure 2: Three Fishburn structures: an interval order, its minimal interval representation, and its Fishburn matrix. In this paper, we use the convention that cells of value 0 in a matrix are depicted as empty boxes. • M = 0 whenever i > j, i,j • ∈ { } for every i 1, . . . , m there is a j such that M > 0, and i,j • ∈ { } for every j 1, . . . , m there is an i such that M > 0. i,j Fishburn matrices were introduced by Fishburn [18] as a convenient way to represent interval orders. Given ≺ × an interval order R = (X, ) of magnitude m with minimal representation I, we represent it by an m m |{ ∈ }| ∈ matrix M such that M is equal to x X; I(x) = [i, j] . We then say that an element x X is i,j represented by the cell (i, j) of M, if I(x) = [i, j]. Thus, every element of R is represented by a unique cell of M, and M is the number of elements of R represented by cell (i, j) of M. This correspondence yields i,j ∈ a bijection between Fishburn matrices and interval orders. For an element x X, we let c denote the cell x of M representing x. Conversely, for a cell c of M, we let X be the set of elements of X represented by c. c Note that the value in cell c is precisely the cardinality of X . c In the rest of this paper, we will use Fishburn matrices as our main family of Fishburn-enumerated object. We now show how the basic features of interval orders translate into matrix terminology. ≺ Observation 1.1. Let R = (X, ) be an interval order, and let M be the corresponding Fishburn matrix. (cid:48) (cid:48) (cid:48) Let x and x be two elements of R, represented respectively by cells (i, j) and (i , j ) of M. The following holds: • The size of R is equal to the weight of M. • The number of minimal elements of R is equal to the weight of the ﬁrst row of M, and the number of maximal elements of R is equal to the weight of the last column of M. • (cid:48) The elements x and x are indistinguishable in R if and only if they are represented by the same cell (cid:48) (cid:48) of M, i.e., i = i and j = j . • (cid:48) ≺ (cid:48) We have x x if and only if j < i. (cid:48) (cid:48) (cid:48) ≤ Let M be a Fishburn matrix, and let c = (i, j) and c = (i , j ) be two cells of M such that i j (cid:48) ≤ (cid:48) and i j , i.e., the two cells are on or above the main diagonal. We shall frequently use the following terminology (see Figure 3): • (cid:48) (cid:48) (cid:48) (cid:48) Cell c is greater than cell c (and c is smaller than c), if j < i. The cells c and c are incomparable if neither of them is greater than the other. These terms are motivated by the last part of Observation 1.1. (cid:48) Note that two cells c and c of M are comparable if and only if the smallest rectangle containing both (cid:48) c and c also contains at least one cell strictly below the main diagonal of M. • (cid:48) (cid:48) (cid:48) Cell c is South of c , if i > i and j = j . North, West and East are deﬁned analogously. Note that in (cid:48) all these cases, the two cells c and c are incomparable. 4 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: f

text: g

text: e

text: a

text: c

text: c

text: d

text: e

text: f

text: b

text: d

text: g

text: a

text: b

text: Figure

text: Three

text: Fishburn

text: structures:

text: an

text: interval

text: order,

text: its

text: minimal

text: interval

text: representation,

text: and

text: its

text: Fishburn

text: matrix.

text: In

text: this

text: paper,

text: we

text: use

text: the

text: convention

text: that

text: cells

text: of

text: value

text: in

text: a

text: matrix

text: are

text: depicted

text: as

text: empty

text: boxes.

text: M

text: whenever

text: i

text: j,

text: i,j

text: for

text: every

text: i

text: m

text: there

text: is

text: a

text: j

text: such

text: that

text: M

text: and

text: i,j

text: for

text: every

text: j

text: m

text: there

text: is

text: an

text: i

text: such

text: that

text: M

text: i,j

text: Fishburn

text: matrices

text: were

text: introduced

text: by

text: Fishburn

text: as

text: a

text: convenient

text: way

text: to

text: represent

text: interval

text: orders.

text: Given

text: an

text: interval

text: order

text: R

text: (X,

text: of

text: magnitude

text: m

text: with

text: minimal

text: representation

text: I,

text: we

text: represent

text: it

text: by

text: an

text: m

text: m

text: matrix

text: M

text: such

text: that

text: M

text: is

text: equal

text: to

text: x

text: X;

text: I(x)

text: [i,

text: j]

text: We

text: then

text: say

text: that

text: an

text: element

text: x

text: X

text: is

text: i,j

text: represented

text: by

text: the

text: cell

text: (i,

text: j)

text: of

text: M,

text: if

text: I(x)

text: [i,

text: j].

text: Thus,

text: every

text: element

text: of

text: R

text: is

text: represented

text: by

text: a

text: unique

text: cell

text: of

text: M,

text: and

text: M

text: is

text: the

text: number

text: of

text: elements

text: of

text: R

text: represented

text: by

text: cell

text: (i,

text: j)

text: of

text: M.

text: This

text: correspondence

text: yields

text: i,j

text: a

text: bijection

text: between

text: Fishburn

text: matrices

text: and

text: interval

text: orders.

text: For

text: an

text: element

text: x

text: X,

text: we

text: let

text: c

text: denote

text: the

text: cell

text: x

text: of

text: M

text: representing

text: x.

text: Conversely,

text: for

text: a

text: cell

text: c

text: of

text: M,

text: we

text: let

text: X

text: be

text: the

text: set

text: of

text: elements

text: of

text: X

text: represented

text: by

text: c.

text: c

text: Note

text: that

text: the

text: value

text: in

text: cell

text: c

text: is

text: precisely

text: the

text: cardinality

text: of

text: X

text: c

text: In

text: the

text: rest

text: of

text: this

text: paper,

text: we

text: will

text: use

text: Fishburn

text: matrices

text: as

text: our

text: main

text: family

text: of

text: Fishburn-enumerated

text: object.

text: We

text: now

text: show

text: how

text: the

text: basic

text: features

text: of

text: interval

text: orders

text: translate

text: into

text: matrix

text: terminology.

text: Observation

text: Let

text: R

text: (X,

text: be

text: an

text: interval

text: order,

text: and

text: let

text: M

text: be

text: the

text: corresponding

text: Fishburn

text: matrix.

text: Let

text: x

text: and

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text: Contents 1 Introduction 1 2 IceCube neutrino events and UHECR data 2 3 Statistical method and data analyses 2 4 Results 6 4.1 Correlations between neutrinos and UHECRs 6 4.2 Energy calibrated events 10 4.3 Astrophysical source search 11 4.4 Neutrino and UHECR luminosities for correlated events 14 5 Discussion and Outlook 18 1 Introduction The IceCube Neutrino Observatory, the world’s largest neutrino detector, has recently pub- ∼ lished neutrino events collected over 3-year period with energy in the 30 TeV–2 PeV range [1]. Shower events, most likely due to ν or ν charge current νN interactions, dominate the e τ event list (28 including 3 events with 1–2 PeV energy) while track events, most likely due to ν charge current νN interactions, constitute the rest. Among a total of 37 events about µ ± 15 could be due to atmospheric neutrino (6.6+−5.9) and muon (8.4 4.2) backgrounds. A 1.6 background-only origin of all 37 events has been rejected at 5.7-σ level [1]. Therefore a cos- ∼ ◦ mic origin of a number of neutrino events is robust. The track events have on average 1 angular resolution, but the dominant, shower events have much poorer angular resolution, ∼ ◦ 15 on average [1], thus making them unsuitable for astronomy. Meanwhile the Pierre Auger Observatory (PAO) [2] and the Telescope Array (TA) [3], two of the world’s largest operating cosmic-ray detectors, have recently released UHECR data collected over more than 10-year and 5-year periods, respectively. Together they have (cid:38) detected 16 events (6 by PAO [2] and 10 by TA [3]) with energies 100 EeV. The total (cid:38) publicly available 100 EeV events including past experiments is 33. While lower-energy cosmic ray arrival directions are scrambled by the Galactic and intergalactic magnetic ﬁelds, (cid:38) at 60 EeV energies the arrival directions of UHECRs tend to be much better correlated with their source directions and astronomy with charged particles could be realized [4]. Few degree angular resolution can be achieved at these energies, which is much better than the IceCube neutrino shower events and is comparable to the neutrino track events. (cid:38) The astrophysical sources of UHECRs with energy 40 EeV need to be located within the so-called GZK volume [5, 6] in order to avoid serious attenuation of ﬂux from them due to interactions of UHECRs with photons from cosmic microwave background (CMB) and extra- galactic background light (EBL). The astrophysical sources of neutrinos, on the other hand, can be located at large distances and still be detected provided their luminosity is suﬃciently high. However, because of weakly interacting nature of neutrinos and limiting luminosity of astrophysical sources, only nearby neutrino sources can be identiﬁed, thus making neutrino astronomy possible. – 1 – ##LTLine## ##LTLine##

text: Contents

text: Introduction

text: IceCube

text: neutrino

text: events

text: and

text: UHECR

text: data

text: Statistical

text: method

text: and

text: data

text: analyses

text: Results

text: Correlations

text: between

text: neutrinos

text: and

text: UHECRs

text: Energy

text: calibrated

text: events

text: Astrophysical

text: source

text: search

text: Neutrino

text: and

text: UHECR

text: luminosities

text: for

text: correlated

text: events

text: Discussion

text: and

text: Outlook

label: section

text: Introduction

text: The

text: IceCube

text: Neutrino

text: Observatory,

text: the

text: world’s

text: largest

text: neutrino

text: detector,

text: has

text: recently

text: pub-

text: lished

text: neutrino

text: events

text: collected

text: over

text: period

text: with

text: energy

text: in

text: the

text: PeV

text: range

text: Shower

text: events,

text: most

text: likely

text: due

text: to

text: ν

text: or

text: ν

text: charge

text: current

text: νN

text: interactions,

text: dominate

text: the

text: e

text: τ

text: event

text: list

text: including

text: events

text: with

text: PeV

text: energy)

text: while

text: track

text: events,

text: most

text: likely

text: due

text: to

text: ν

text: charge

text: current

text: νN

text: interactions,

text: constitute

text: the

text: rest.

text: Among

text: a

text: total

text: of

text: events

text: about

text: µ

text: could

text: be

text: due

text: to

text: atmospheric

text: neutrino

text: and

text: muon

text: backgrounds.

text: A

text: background-only

text: origin

text: of

text: all

text: events

text: has

text: been

text: rejected

text: at

text: level

text: Therefore

text: a

text: cos-

text: mic

text: origin

text: of

text: a

text: number

text: of

text: neutrino

text: events

text: is

text: robust.

text: The

text: track

text: events

text: have

text: on

text: average

text: angular

text: resolution,

text: but

text: the

text: dominant,

text: shower

text: events

text: have

text: much

text: poorer

text: angular

text: resolution,

text: on

text: average

text: thus

text: making

text: them

text: unsuitable

text: for

text: astronomy.

text: Meanwhile

text: the

text: Pierre

text: Auger

text: Observatory

text: (PAO)

text: and

text: the

text: Telescope

text: Array

text: (TA)

text: two

text: of

text: the

text: world’s

text: largest

text: operating

text: cosmic-ray

text: detectors,

text: have

text: recently

text: released

text: UHECR

text: data

text: collected

text: over

text: more

text: than

text: and

text: periods,

text: respectively.

text: Together

text: they

text: have

text: detected

text: events

text: by

text: PAO

text: and

text: by

text: TA

text: with

text: energies

text: EeV.

text: The

text: total

text: publicly

text: available

text: EeV

text: events

text: including

text: past

text: experiments

text: is

text: While

text: lower-energy

text: cosmic

text: ray

text: arrival

text: directions

text: are

text: scrambled

text: by

text: the

text: Galactic

text: and

text: intergalactic

text: magnetic

text: ﬁelds,

text: at

text: EeV

text: energies

text: the

text: arrival

text: directions

text: of

text: UHECRs

text: tend

text: to

text: be

text: much

text: better

text: correlated

text: with

text: their

text: source

text: directions

text: and

text: astronomy

text: with

text: charged

text: particles

text: could

text: be

text: realized

text: Few

text: degree

text: angular

text: resolution

text: can

text: be

text: achieved

text: at

text: these

text: energies,

text: which

text: is

text: much

text: better

text: than

text: the

text: IceCube

text: neutrino

text: shower

text: events

text: and

text: is

text: comparable

text: to

text: the

text: neutrino

text: track

text: events.

text: The

text: astrophysical

text: sources

text: of

text: UHECRs

text: with

text: energy

text: EeV

text: need

text: to

text: be

text: located

text: within

text: the

text: so-called

text: GZK

text: volume

text: in

text: order

text: to

text: avoid

text: serious

text: attenuation

text: of

text: ﬂux

text: from

text: them

text: due

text: to

text: interactions

text: of

text: UHECRs

text: with

text: photons

text: from

text: cosmic

text: microwave

text: background

text: (CMB)

text: and

text: extra-

text: galactic

text: background

text: light

text: (EBL).

text: The

text: astrophysical

text: sources

text: of

text: neutrinos,

text: on

text: the

text: other

text: hand,

text: can

text: be

text: located

text: at

text: large

text: distances

text: and

text: still

text: be

text: detected

text: provided

text: their

text: luminosity

text: is

text: suﬃciently

text: high.

text: However,

text: because

text: of

text: weakly

text: interacting

text: nature

text: of

text: neutrinos

text: and

text: limiting

text: luminosity

text: of

text: astrophysical

text: sources,

text: only

text: nearby

text: neutrino

text: sources

text: can

text: be

text: identiﬁed,

text: thus

text: making

text: neutrino

text: astronomy

text: possible.

text: ##LTLine##

text: ##LTLine##


text: stnuoC stnuoC stnuoC stnuoC PAO > 80 EeV TA > 80 EeV 16 16 14 P-value= 0.127 14 P-value=0.064 12 P-value=0.159 12 P-value=0.095 10 10 8 8 6 6 4 4 2 2 0 0 0 2 4 6 8 10 0 2 4 6 8 10 δχ2 δχ2 TA+PAO > 80 EeV TA+PAO+AGASA > 80 EeV 16 16 14 P-value=0.13 14 P-value=0.267 12 P-value=0.197 12 P-value=0.32 10 10 8 8 6 6 4 4 2 2 0 0 0 2 4 6 8 10 0 2 4 6 8 10 δχ2 δχ2 ≥ Figure 5. The same as in Figure 3 but for UHECRs with energy 80 EeV. (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) − ≈ ◦ E 1 B D 1/2 λ 1/2 δθ 1.1 Z cr rdm coh (4.1) IG 100 EeV 1 nG 200 Mpc 100 kpc where Z and E are the charge and energy of the UHECR and D is the distance to the cr source. The deﬂection angle in a small-scale Galactic random magnetic ﬁeld, using Eq. (4.1) ≈ ◦ with B = 1µG, λ = 100 pc is much smaller, δθ 0.2 Z, for E = 100 EeV and rdm coh G cr D = 10 kpc. However, the deﬂection angle in the large-scale regular component of the ∼ ◦ ◦ Galactic magnetic ﬁeld in the disk and in the halo can be larger, 1 –3 [19, 20]. Hereafter ≥ we assume that UHECRs with energy 80 EeV are dominantly protons 1. We also chose a 1Note that the mass composition measurement by the PAO collaboration by using shower maxima, which ∼ favors heavy nuclei as primaries, is done up to an energy 60 EeV only [22]. – 11 – ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: stnuoC

text: stnuoC

text: stnuoC

text: stnuoC

text: PAO

text: EeV

text: TA

text: EeV

text: P-value=

text: δχ2

text: δχ2

text: TA+PAO

text: EeV

text: TA+PAO+AGASA

text: EeV

text: δχ2

text: δχ2

text: Figure

text: The

text: same

text: as

text: in

text: Figure

text: but

text: for

text: UHECRs

text: with

text: energy

text: EeV.

text: E

text: B

text: D

text: λ

text: δθ

text: Z

text: cr

text: rdm

text: coh

text: IG

text: EeV

text: nG

text: Mpc

text: kpc

text: where

text: Z

text: and

text: E

text: are

text: the

text: charge

text: and

text: energy

text: of

text: the

text: UHECR

text: and

text: D

text: is

text: the

text: distance

text: to

text: the

text: cr

text: source.

text: The

text: deﬂection

text: angle

text: in

text: a

text: small-scale

text: Galactic

text: random

text: magnetic

text: ﬁeld,

text: using

text: Eq.

text: with

text: B

text: λ

text: pc

text: is

text: much

text: smaller,

text: δθ

text: Z,

text: for

text: E

text: EeV

text: and

text: rdm

text: coh

text: G

text: cr

text: D

text: kpc.

text: However,

text: the

text: deﬂection

text: angle

text: in

text: the

text: large-scale

text: regular

text: component

text: of

text: the

text: Galactic

text: magnetic

text: ﬁeld

text: in

text: the

text: disk

text: and

text: in

text: the

text: halo

text: can

text: be

text: larger,

text: –3

text: Hereafter

text: we

text: assume

text: that

text: UHECRs

text: with

text: energy

text: EeV

text: are

text: dominantly

text: protons

text: We

text: also

text: chose

text: a

text: that

text: the

text: mass

text: composition

text: measurement

text: by

text: the

text: PAO

text: collaboration

text: by

text: using

text: shower

text: maxima,

text: which

text: favors

text: heavy

text: nuclei

text: as

text: primaries,

text: is

text: done

text: up

text: to

text: an

text: energy

text: EeV

text: only

text: ##LTFigure##

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