## 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.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 Γ and G, where Γ 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 Γ is a complete or null graph or a tree.

The paper concludes with some comments on other polynomials associated with a permutation group.

for the number of elements of G with a given number of cycles.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 Γ and G, where Γ 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 Γ is a complete or null graph or a tree.

The paper concludes with some comments on other polynomials associated with a permutation group.

Original language | English |
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Article number | P1.14 |

Number of pages | 13 |

Journal | Electronic Journal of Combinatorics |

Volume | 25 |

Issue number | 1 |

Publication status | Published - 25 Jan 2018 |

## Keywords

- Permutation group
- Chromatic polynomial
- Reciprocity