On a generalization of index series of species
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 \(\Gamma_ F\) and \(Z_ F\) by means of some \(N_ M\)'s \((M, N\in {\mathcal M})\), illustrated by examples dealing with molecular species \(F\) on 4 points. For transitive species additional results are proved. Possible directions for further research are stated.
scientific article; zbMATH DE number 720824
scientific article