The cycle polynomial of a permutation group
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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.
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 >
Electronic Journal of Combinatorics
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