A generalized Mahonian statistic on absorption ring mappings
type: Item
instance of: scholarly article
title: A generalized Mahonian statistic on absorption ring mappings
zbMATH Open document ID: 0883.05004
author: Don Rawlings
published in: Journal of Combinatorial Theory. Series A
publication date: 1998-02-23T00:00:00Z
full work available at URL: https://digitalcommons.calpoly.edu/cgi/viewcontent.cgi
review text: The article under review describes and discusses a statistic on a new combinatorial object, absorption ring mappings. Suppose we are given a \(k\)-tuple of sequences of distinct integers together with a permutation \(\sigma \in S_n\). An absorption ring mapping is obtained by inserting\ those sequences into a circular arrangement of cells (or ring) which is labelled according to \(\sigma \). The actual insertion process begins by choosing an initial placement for the first sequence. The final placement of the first (and subsequent) sequences is determined by a coin-flipping scheme. In the end, the process determines a (not necessarily one-to-one) function from the set of integers which occur in the given sequences to the set of labels, \(\{1,\ldots ,n\}\). The number of tails in the shortest coin-flipping sequence required to generate the function is, therefore, a statistic on the collection of all such functions (absorption ring mappings). Set \textbf{J} to be the given \(k\)-tuple of sequences, \(j_1,\dots,j_k\) the lengths of those sequences, \(AR_n({\mathbf J})\) the collection of absorption ring mappings generated as described above, and, for \(f\in AR_n({\mathbf J})\), let \(|f|\) denote the number of tails in a minimal flipping sequence. The key result gives the distribution generating function for this statistic, that is, \[ \sum_fq^{|f|}=[n][n-j_1]\cdots [n-j_1-j_2-\cdots -j_k] \] irrespective of choice of initial placement rule or choice of \(\sigma \). The proof follows easily from the definitions and the independence of Bernoulli trials. In case \(k=n\), so that each sequence has just one term, the collection of absorption ring mappings is \(S_n\) and the statistic \(|\;|\) reduces to the generalized Mahonian statistic \(den^a\) described by \textit{G.-H. Han} [``Calcul Denertien'', doctoral thesis, IRMA, Univ. de Strasbourg, 1991]. Moreover, with only modest modifications, \(|\;|\) can be viewed as a statistic on multiset permutations; again, it is equi…
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A05
zbMATH DE Number: 1051480
zbMATH Keywords: permutation statistics
zbMATH Keywords: Mahonian statistics
reviewed by: Jennifer R.Galovich
MaRDI profile type: Publication
OpenAlex ID: W1987728021
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DOI: 10.1006/JCTA.1997.2783
description: scientific article; zbMATH DE number 1051480