Order Polynomials and Pólya's Enumeration Theorem

Pólya’s enumeration theorem states that the number of labelings of a finite set up to symmetry is given by a polynomial in the number of labels. We give a new perspective on this theorem by generalizing it to partially ordered sets and order preserving maps. Further we prove a reciprocity statement in terms of strictly order preserving maps generalizing a classical result by Stanley (1970). We apply our results to counting graph colorings up to symmetry.

Why Are There Twenty-Nine Tetrachords? A Tutorial on Combinatorics and Enumeration in Music Theory

Many techniques in combinatorial mathematics have applications in music theory. Standard formulas for permutations and combinations may be used to enumerate melodies, rhythms, rows, pitch-class sets, and other familiar musical entities subject to various constraints on their structure. Some music scholars in the eighteenth century advocated elementary combinatorial methods, including dice games, as aids in composition. Problems involving the enumeration of set classes, row classes, and other types of equivalence classes are more difficult and require advanced techniques for their solution, notably Pólya’s Enumeration Theorem. Such techniques are applicable in a wide variety of situations, enabling the enumeration of diverse musical structures in scales of various cardinalities and under various definitions of equivalence relations.

An infinite version of the Pólya enumeration theorem

Using measure theory, the orbit counting form of Pólya's enumeration theorem is extended to countably infinite discrete groups.