A \(D\)-finiteness result for products of permutations
Let \(e_ k^{\mathbf p}({\mathbf n})\), \({\mathbf p}=(p_ 1,\dots,p_ d)\) and \({\mathbf n}=(n_ 1,\dots,n_ d)\), denote the number of permutations of \(N=n_ 1p_ 1+\cdots+n_ dp_ d\) that can be represented as the product of an arbitrary but fixed full cycle and a permutation with \(n_ i\) cycles of length \(p_ i\), \(1\leq i\leq d\). In ``Counting cycles in permutations by group characters, with an application to a topological problem'' [Trans. Am. Math. Soc. 299, 785-801 (1987; Zbl 0655.05005)], the second author found a closed expression for the generating function for \(e_ k^{\mathbf p}({\mathbf n})\). In this paper, the generating function \(\sum_{k,{\mathbf n}}e_ k^{\mathbf p}({\mathbf n}) z^ kx^ N{\mathbf w}^{\mathbf n}\) is proven to be \(D\)-finite in the variables \(x\) and \({\mathbf w}\) over the field \({\mathbf Q}(z)\). This means that the vector space spanned by all partial derivatives of this series is finite dimensional over \({\mathbf Q}(z)\), and thus the counting function satisfies a linear recurrence with a fixed number of terms. The authors also find an explicit formula for \(e_ k^{\mathbf p}({\mathbf n})\) in terms of the characters associated with the irreducible representations indexed by the conjugacy classes associated with hooks (partitions of the form \(1^ r\), \(N-r)\) evaluated on the conjugacy class indexed by \(p_ 1^{n_ 1},\dots,p_ d^{n_ d}\).
scientific article; zbMATH DE number 63150
scientific article