TITLE: Cyclotomic Polynomials in Combinatorics
QUESTION [15 upvotes]: I am searching for a combinatorial significance of cyclotomic polynomials. The only examples I got are a paper by Neville Robbins http://www.emis.de/journals/INTEGERS/papers/a6/a6.pdf and two recent papers by Gregg Musiker and Victor Reiner http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.1844v2.pdf and http://combinatorics.cis.strath.ac.uk/fpsac2011/proceedings/dmAO0161.pdf but these do not essentially give a clear picture. I am interested in examples that relate cyclotomic polynomials to foundations of combinatorics (if these exist) or if someone can give a direct combinatorial interpretation of coefficients of cylotomic polynomials that'll be quite helpful.

REPLY [4 votes]: A recent paper of mine on permutation enumeration that uses cyclotomic polynomials is I. M. Gessel, Reciprocals of exponential polynomials and permutation enumeration, Australasian J. Combin. 74 (2) (2019), 364–370.