Cyclotomic Polynomials in Combinatorics 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.
 A: I don't know that this is foundational, but see
http://en.wikipedia.org/wiki/Sicherman_dice
A: The following may be helpful to you:
Coefficients of Cyclotomic Polynomials - Jordan Bell
Memoirs of the AMS - Number 510 - On the Coefficients of Cyclotomic Polynomials - Gennady Bachman
Combinatorial Aspects of Elliptic Curves
A: 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.
A: Ethan Coven and Aaron Meyerowitz have a paper "Tiling the integers with translates of one finite set" (cited recently on Terry Tao's blog) about tiling the integers with a single prototile. 
That is, they are looking for conditions under which subsets of the integers, $A$ say, such that there are $n_1,n_2,n_3,...$ such that $n_1+A$, $n_2+A$, $\ldots$ disjointly cover all of the integers.
In Coven and Meyerowitz's paper, two conditions are given directly in terms of cyclotomic polynomials for $A$ to have this property. If both conditions are satisfied, then $A$ tiles the integers. On the other hand, if $A$ tiles the integers, then one of the conditions is shown to be satisfied. They conjecture that the second condition must also be satisfied.
A: http://math.ecnu.edu.cn/~jwguo/maths/congruence.pdf
Here, the cyclotomic polynomials appear in an identity involving q-binomial coefficients.
A: My first thought was the pair of papers by Musiker and Reiner that you mention.  Another paper involving combinatorics and cyclotomic polynomials is the following.
Cyclotomic factors of the descent set polynomial
Denis Chebikin, Richard Ehrenborg, Pavlo Pylyavskyy, Margaret Readdy
Journal of Combinatorial Theory, Series A
Volume 116, Issue 2, February 2009, Pages 247-264
A: I remember seeing in my student days in constructions of Block Designs, or more specifically, mutually orthogonal latin squares, finite fields play a crucial role and cyclotomic polynomials enter there. Sorry I cannot be any more specific.
