On a generalization of index series of species
type: Item
instance of: scholarly article
title: On a generalization of index series of species
zbMATH Open document ID: 0821.05060
DOI: 10.1016/0097-3165(95)90104-3
author: Gilbert Labelle
author: Jacques Labelle
author: Kathleen Pineau
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-09-12T00:00:00Z
review text: This paper presents a common generalization of the following two notions: the cycle indicator series \(Z_ F\) and the asymmetry indicator series \(\Gamma_ F\) of a species \(F\) of structures. The symmetry types of \(F\) are coded by a species \(G\). Then the \(G\)-indicator series of \(F\) (\(G\)- série indicatrice de \(F\)), denoted by \(G_ F\), is defined as follows: (1) If \(M\), \(N\) are molecular species (i.e., there is no non-trivial decomposition into a sum of species) then one can write \(M= X^ m/H\), \(N= X^ n/K\) with subgroups \(H\subseteq S_ m\), \(K\subseteq S_ n\) (\(H\) is the stabilizer of an \(M\)-structure with label set \(\{1,\dots, m\}\); analogously for \(K\)), and \(N_ M(x_ 1, x_ 2,\dots)\), the \(N\)- indicator series of \(M\), expressed in the base of power sums \(x_ i= \sum_{k\geq 1} t^ i_ k\) is---roughly spoken---defined to be the sum of the weights \(t^{n_ 1(s)}_ 1 t^{n_ 2(s)}_ 2\dots\) of all types of the \(M_ t(X)\)-structure \(s\) the stabilizer of which is conjugated to \(K\). (If \(m\neq n\) then \(N_ M= 0\).) In case that \(m= n\), using the Möbius function on the lattice of the subgroups of \(H\), there is proved a formula representing \(N_ M(x_ 1, x_ 2,\dots)\) as a power series in \(x_ 1, x_ 2,\dots\) effecting the enumeration of the structures of \(M\) according to the subgroups of \(H\) conjugated to \(K\). (2) Let \(\mathcal M\) denote the set of all molecular species and \[ F= \sum_{M\in {\mathcal M}} f_ M M\quad\text{and}\quad G= \sum_{N\in {\mathcal M}} g_ N N \] be the unique molecular decompositions of the species \(F\) and \(G\), respectively. Then \[ G_ F(x_ 1, x_ 2,\dots):= \sum_{M, N\in {\mathcal M}} g_ N f_ M N_ M (x_ 1, x_ 2,\dots). \] By suitable choice of \(G\), namely \(G= L= \sum_{n\geq 0} X^ n=\) the species of linear orders and \(G= Z= \sum_{N\in {\mathcal M}} N\), the \(G\)-indicator series of \(F\) is proved to reduce to the series \(\Gamma_ F\) and \(Z_ F\), respectively. This enables the calculation of \…
reviewed by: Günter Schaar
Mathematics Subject Classification ID: 05E99
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 720824
zbMATH Keywords: index series
zbMATH Keywords: generating function
zbMATH Keywords: cycle indicator series
zbMATH Keywords: asymmetry indicator series
zbMATH Keywords: species
zbMATH Keywords: molecular species
zbMATH Keywords: transitive species
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90104-3
OpenAlex ID: W2204051287
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description: scientific article; zbMATH DE number 720824