Permutation enumeration of the symmetric group and the combinatorics of symmetric functions
type: Item
instance of: scholarly article
title: Permutation enumeration of the symmetric group and the combinatorics of symmetric functions
zbMATH Open document ID: 0833.05078
DOI: 10.1016/0097-3165(95)90027-6
author: Desiree A. Beck
author: Jeffrey Remmel
published in: Journal of Combinatorial Theory. Series A
publication date: 1996-03-05T00:00:00Z
review text: \textit{F. Brenti} [Pac. J. Math. 157, No. 1, 1-28 (1993; Zbl 0805.05089)] has shown that the polynomials obtained by enumerating conjugacy classes of the symmetric group \(S_n\) with respect to excedances are unimodal and symmetric, and thus arises naturally from the theory of symmetric functions. The authors give combinatorial proofs and interpretations for the various bases of symmetric functions under \(\xi\), a homomorphism (introduced by Brenti) on the ring of symmetric functions, where \(\xi(e_i)= (1- x)^{k- 1}/k!\) is the image of the elementary symmetric function \(e_k\), \(\xi(e_0)= 1\). Their combinatorial methods lead to numerous extensions, e.g. \(q\)-analogues which involve the inversion statistics, as well as a definition of new combinatorial objects---border rim hook tabloids. Numerous examples are included.
reviewed by: Indulis Strazdins
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 20B30
zbMATH DE Number: 802348
zbMATH Keywords: permutation enumeration
zbMATH Keywords: symmetric group
zbMATH Keywords: unimodal
zbMATH Keywords: symmetric functions
zbMATH Keywords: inversion statistics
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90027-6
OpenAlex ID: W2070408914
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description: scientific article; zbMATH DE number 802348