Cyclic Sieving for Strong Dichotomy Enumeration

TL;DR

This paper proves a conjecture relating to the enumeration of strong dichotomy classes in cyclic groups, confirming a sign pattern in the rigid pattern-inventory polynomial for all odd k.

Contribution

It generalizes previous results by proving the conjecture for all odd k, expanding understanding of cyclic sieving in combinatorial enumeration.

Findings

Confirmed the conjecture for all odd k.

Established the sign pattern of the polynomial evaluation at -1.

Connected enumeration with algebraic group actions.

Abstract

Agust\'{i}n-Aquino solved, in terms of the table of marks of $\Aff(\mathbb{Z}/2k\mathbb{Z})$, the problem of enumerating the classes of bicolour self-complementary and rigid patterns in $\mathbb{Z}/2k\mathbb{Z}$ (also known as \emph{strong dichotomy classes}). In particular, the rigid pattern-inventory polynomial appeared, for odd $k$, to yield the number of strong classes with negative sign when evaluated in $-1$, and it was conjectured that this is true for $k$ a power of an odd prime. Here we prove the conjecture is true for $k$ odd in general.