Problems on the Z-statistic
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.
scientific article; zbMATH DE number 4108769
scientific article