Scrambled Vandermonde convolutions of Gaussian polynomials

Abstract

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