Gandhi \(q\)-polynomials and Denert statistic
type: Item
instance of: scholarly article
title: Gandhi \(q\)-polynomials and Denert statistic
zbMATH Open document ID: 0944.05008
DOI: 10.1016/S0012-365X(97)00189-1
author: Guo-Niu Han
author: Jiang Zeng
published in: Discrete Mathematics
publication date: 2000-09-24T00:00:00Z
review text: The authors use Hahn's \(q\)-difference operator to derive \(q\)-analogues of the Gandhi polynomials as \(y\) specializations of certain polynomials denoted \(D_n(x,y,q)\). They give a continued fraction expansion for the generating function of the polynomials \(D_n(x,y,q)\), and, inspired by the work of Dumont in this area, they also construct two different combinatorial interpretations of \(D_n(x,y,q)\) as a trivariate generating function on \(G_n\), the set of Genocchi permutations. The first interpretation is based on an intermediate interpretation using the set of Dyck paths with \(2n+ 1\) steps. The second is based on a statistic-preserving bijection between \(G_n\) and the set \(P_n\) of ``pistolets'', maps \(f\) from \([2n]\) onto \(2[n]\) with the property that \(i\leq f(i)\) for all \(i\). In both cases a key ingredient is the Denert statistic (as described by D. Foata and D. Zeilberger). The construction of an involution on \(G_n\) which directly relates the two trivariate statistics remains open.
reviewed by: Jennifer R.Galovich
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11B65
zbMATH DE Number: 1334682
zbMATH Keywords: Gandhi polynomials
zbMATH Keywords: continued fraction expansion
zbMATH Keywords: generating function
zbMATH Keywords: Genocchi permutations
zbMATH Keywords: Dyck paths
zbMATH Keywords: Denert statistic
MaRDI profile type: Publication
full work available at URL: s0012-365x(97)00189-1
OpenAlex ID: W2077511480
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description: scientific article; zbMATH DE number 1334682