The cycle polynomial of a permutation group

Peter J. Cameron; Jason Semeraro

The cycle polynomial of a permutation group

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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.

Original language	English
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

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AB - 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.

KW - Permutation group

KW - Chromatic polynomial

KW - Reciprocity

UR - https://www.scopus.com/pages/publications/85042212781

M3 - Article

SN - 1077-8926

VL - 25

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 1

M1 - P1.14

ER -

The cycle polynomial of a permutation group

Abstract

Keywords

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