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dc.contributor.authorCameron, Peter J.
dc.contributor.authorSemeraro, Jason
dc.identifier.citationCameron , P J & Semeraro , J 2018 , ' The cycle polynomial of a permutation group ' , Electronic Journal of Combinatorics , vol. 25 , no. 1 , P1.14 . < >en
dc.identifier.otherPURE: 252130100
dc.identifier.otherPURE UUID: 9c005015-45e4-4405-9166-1461765e8204
dc.identifier.otherScopus: 85042212781
dc.identifier.otherORCID: /0000-0003-3130-9505/work/58055707
dc.identifier.otherWOS: 000432156200009
dc.description.abstractThe 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.
dc.relation.ispartofElectronic Journal of Combinatoricsen
dc.rightsCopyright (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
dc.subjectPermutation groupen
dc.subjectChromatic polynomialen
dc.subjectQA Mathematicsen
dc.titleThe cycle polynomial of a permutation groupen
dc.typeJournal articleen
dc.description.versionPublisher PDFen
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews.Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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