Scrambled Vandermonde convolutions of Gaussian polynomials

Abstract

It is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the ▪ statistic on monotone paths in a rectangular grid. We introduce two new statistics, ▪ and ▪; attach “ornaments” to the grid that scramble the values of ▪ in specific fashion; and re-evaluate these statistics, in order to argue that all scrambled versions of the ▪ statistic are equidistributed with ▪. Our main result is a representation of the generating function for the bi-statistic ▪ as a new, two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.