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The cycle polynomial of a permutation group

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Cameron_Combinatorics_Cycle_polynominal_VoR.pdf (270.3Kb)
Date
25/01/2018
Author
Cameron, Peter J.
Semeraro, Jason
Keywords
Permutation group
Chromatic polynomial
Reciprocity
QA Mathematics
Mathematics(all)
NDAS
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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.
Citation
Cameron , P J & Semeraro , J 2018 , ' The cycle polynomial of a permutation group ' , Electronic Journal of Combinatorics , vol. 25 , no. 1 , P1.14 . < http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p14 >
Publication
Electronic Journal of Combinatorics
Status
Peer reviewed
ISSN
1077-8926
Type
Journal article
Rights
Copyright (c)2017 the authors. This work is made available online in accordance with the publisher’s policies. This is the final published version of the work which was originally published at http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p14
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  • University of St Andrews Research
URL
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p14
URI
http://hdl.handle.net/10023/12840

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