Problems on the Z-statistic
type: Item
instance of: scholarly article
title: Problems on the Z-statistic
zbMATH Open document ID: 0677.05007
DOI: 10.1016/0012-365X(88)90131-8
published in: Discrete Mathematics
publication date: 1989-01-01T00:00:00Z
review text: There exist two similar proofs of Andrews' q-Dyson conjecture \[ [x^ 0]\prod_{1\leq i<j\leq n}\{\prod^{a_ i}_{k=1}(1-\frac{x_ i}{x_ j}q^{k-1})\cdot \prod^{a_ j}_{k=1}(-\frac{x_ j}{x_ i}q^ k)\} \] \[ =\frac{(q)_{a_ 1+a_ 2+...+a_ n}}{(q)_{a_ 1}(q)_{a_ 2}...(q)_{a_ n}}=\begin{bmatrix} a_ 1+...+a_ n\\ a_ 1,...,a_ n\end{bmatrix}, \] where \([x^{\alpha}]\) denotes the coefficients of \(x^{\alpha}\) in the expressions that follow and \((q)_ a=(1-q)(1-q^ 2)...(1-q^ a)\). These proofs rely on showing that both sides are generating functions for a statistic on words which was introduced by Zeilberger (called Z-statistic). This paper is a collection of observations and questions that have arisen in connection with the Z- statistic and ten problems are presented which have arisen in the attempt to understand the Z-statistic.
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 4108769
zbMATH Keywords: root system
zbMATH Keywords: q-Dyson conjecture
zbMATH Keywords: Z-statistic
author: David M. Bressoud
MaRDI profile type: Publication
cites work: On the Geometry of Numbers in Elementary Number Theory
cites work: Constant Term Identities Extending the q-Dyson Theorem
cites work: On the Netto Inversion Number of a Sequence
cites work: A Rogers-Ramanujan bijection
cites work: Q3731609
cites work: A Proof of Askey’s Conjectured q-Analogue of Selberg’s Integral and a Conjecture of Morris
cites work: Some Conjectures for Root Systems
cites work: A Proof of the $G_2 $ Case of Macdonald’s Root System-Dyson Conjecture
cites work: A proof of Andrews' \(q\)-Dyson conjecture
full work available at URL: 0012-365x(88)90131-8
OpenAlex ID: W1991627383
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description: scientific article; zbMATH DE number 4108769