TITLE: Counting non-isomorphic relations
QUESTION [12 upvotes]: On a set $X$ of $n$ elements, how many non-isomorphic relations are there? The number of relations on a set of $n$ elements is $|\mathcal{P}(X \times X)|=2^{n^2}$, but is there any way to give a general formula for how many of these are non-isomorphic? I am interested in this question for an application to modal logic. Thanks!
In particular, I consider two relations $R$ and $S$ to be isomorphic if there exists a bijection $\varphi: X \rightarrow X$ such that $\varphi(x) R \varphi(y) \Leftrightarrow x S y$

REPLY [8 votes]: Here are some more details  concerning this computation.  We solve the
problem using the Polya Enumeration Theorem, which actually gives more
information, permitting  the classification of these  relations by the
number of edges.  This computation is very similar  to the enumeration
of      non-isomorphic     graphs      shown     at      this     MSE
link.
We need  to compute the cycle  index of the edge  permutation group of
the complete bipartite graph on $2n$ nodes. This is almost the same as
the  edge  permutation  group  of  a graph  on  $n$  nodes.  The  only
difference is that  the appropriate group is obtained  from the action
of the  symmetric group $S_n$  on ordered pairs rather  than unordered
pairs as in  the edge permutation group. This  actually simplifies the
analysis because  a pair of vertices  on an even cycle  does not close
the cycle after traversing $n/2$  elements but rather traverses all of
the cycle.
With  the cycle  index we  are  free to  apply PET  or Burnside.   The
following Maple code computes this cycle index from the cycle index of
the symmetric group.

pet_cycleind_symm :=
proc(n)
option remember;

    if n=0 then return 1; fi;

    expand(1/n*
           add(a[l]*pet_cycleind_symm(n-l), l=1..n));
end;

pet_varinto_cind :=
proc(poly, ind)
local subs1, subsl, polyvars, indvars, v, pot;

    polyvars := indets(poly);
    indvars := indets(ind);

    subsl := [];

    for v in indvars do
        pot := op(1, v);

        subs1 :=
        [seq(polyvars[k]=polyvars[k]^pot,
             k=1..nops(polyvars))];

        subsl := [op(subsl), v=subs(subs1, poly)];
    od;

    subs(subsl, ind);
end;

pet_cycleind_rel :=
proc(n)
option remember;
local dsjc, flat, p, cyc1, cyc2, l1, l2, res;

    if n=0 then return 1; fi;
    if n=1 then return a[1] fi;

    res := 0;

    for dsjc in pet_cycleind_symm(n) do

        p := 1;

        for cyc1 in indets(dsjc) do
            l1 := op(1, cyc1);
            for cyc2 in indets(dsjc) do
                l2 := op(1, cyc2);

                p := p *
                a[lcm(l1,l2)] ^
                (degree(dsjc, cyc1)*degree(dsjc, cyc2) *
                 l1*l2/lcm(l1, l2));
            od;
        od;

        res := res + lcoeff(dsjc)*p;
    od;

    res;
end;


gf :=
proc(n)
option remember;
local gf;

    gf := pet_varinto_cind(1+z, pet_cycleind_rel(n));
    expand(gf);
end;

This is the cycle index for $n=3$:
$$1/6\,{a_{{1}}}^{9}+1/2\,{a_{{2}}}^{4}a_{{1}}+1/3\,{a_{{3}}}^{3}$$
and here it is for $n=4$:
$$1/24\,{a_{{1}}}^{16}+1/4\,{a_{{1}}}^{4}{a_{{2}}}^{6}
+1/3\,{a_{{3}}}^{5}a_{{1}}+1/8\,
{a_{{2}}}^{8}+1/4\,{a_{{4}}}^{4}. $$
The substituted cycle index for $n=4$ is
$${z}^{16}+2\,{z}^{15}+9\,{z}^{14}+32\,{z}^{13}+95\,{z}^{12}+203\,{z}^{11}
+373\,{z}^{10}+515\,{z}^{9}\\+584\,{z}^{8}+515\,{z}^{7}+373\,{z}^{6}
+203\,{z}^{5}+95\,{z}^{4}+32\,{z}^{3}+9\,{z}^{2}+2\,z+1 .$$
This shows that  there are e.g. $515$  non-isomorphic relations having
$7$ edges when $n=4.$
For $n=5$ we get
$$ {\frac {1}{120}}\,{a_{{1}}}^{25}+1/12\,{a_{{1}}}^{9}{a_{{2}}}^{8}
+1/6\,{a_{{3}}}^{7}{a_{{1}}}^{4}+1/8\,{a_{{2}}}^{12}a_{{1}}
\\+1/4\,{a_{{4}}}^{6}a_{{1}}+1/6\,{a_{{2}}}^{2
}{a_{{6}}}^{2}{a_{{3}}}^{3}+1/5\,{a_{{5}}}^{5}.$$
The count of the number of non-isomorphic binary relations is
$$2, 10, 104, 3044, 291968, 96928992, 112282908928, 458297100061728,\\
6666621572153927936,
349390545493499839161856,\\66603421985078180758538636288,\\
46557456482586989066031126651104256,
\\ 120168591267113007604119117625289606148096,\\
1152050155760474157553893461743236772303142428672,\\
41233441401686067929518834693764864820973598636585435136,\ldots$$
This is indeed OEIS A000595 as noted by the
other poster.  The count does not  require PET and can  be obtained by
Burnside as shown below.

Q :=
proc(n)
option remember;
local cind, vars;

    cind := pet_cycleind_rel(n);
    vars := indets(cind);

    subs({seq(v = 2, v in vars)}, cind);
end;

Note  that the  bipartite graph  here is  special, consisting  of $2n$
nodes with $n$ on  the left and $n$ on the right,  both labeled $1$ to
$n$.  The cycle  index  represents  the action  on  the  edges by  the
simultaneous action  of the symmetric group  $S_n$ on the two  sets of
labels.