# The cycle polynomial of a permutation group

## TL;DR

This paper studies the cycle polynomial of permutation groups, explores its properties, and investigates reciprocal relations with other combinatorial polynomials, providing examples and characterizations.

## Contribution

It introduces the concept of the cycle polynomial, examines its properties, and establishes connections with reciprocal polynomials like the orbital chromatic polynomial, including specific cases and open problems.

## Key findings

- Cycle polynomial properties are characterized and exemplified.
- Reciprocal relations between cycle polynomial and orbital chromatic polynomial are identified.
- Examples include complete graphs, null graphs, and trees.

## Abstract

The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles: \[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $\Omega$. In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples. In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs of combinatorial polynomials. We show that, in a considerable number of cases, there is a polynomial in the reciprocal relation to the cycle polynomial of $G$; this is the orbital chromatic polynomial of $\Gamma$ and $G$, where $\Gamma$ is a $G$-invariant graph, introduced by the first author, Jackson and Rudd. We pose the general problem of finding all such reciprocal pairs, and give a number of examples and characterisations: the latter include the cases where $\Gamma$ is a complete or null graph or a tree. The paper concludes with some comments on other polynomials associated with a permutation group.