Abstract
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.
Highlights
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
Counting objects up to symmetry is a basic problem of enumerative combinatorics
Summary
Counting objects up to symmetry is a basic problem of enumerative combinatorics. A fundamental result in this context is Polya’s enumeration theorem which is concerned with counting labelings of a set of objects modulo symmetry. Further we give a combinatorial interpretation for ΩP,G(−n) in terms of orbits of strictly order preserving maps. This naturally generalizes the classical polynomiality and reciprocity theorems for order preserving maps due to Stanley [4]. These results can be generalized to counting (P, ω)-partitions up to symmetry. We give a combinatorial interpretation for evaluating this polynomial at negative integers in terms of acyclic orientations and compatible colorings This naturally generalizes Stanley’s reciprocity for graph colorings [6]
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