Scrambled Vandermonde convolutions of Gaussian polynomials

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.