Abstract: Alternating sign triangles (ASTs) have been introduced by Ayyer, Behrend and Fischer in 2016 and it was proven that there is the same number of ASTs with $n$-rows as there is of $n\times n$ alternating sign matrices (ASMs). Later on a refined enumeration of ASTs with respect to a statistic $\rho$, having the same distribution as the unique 1 in the top row of an ASM, was given by connecting ASTs to $(0,n,n)$-Magog-trapezoids. We introduce two more statistics counting the all $0$-columns on the left and right in an AST yielding objects we call alternating sign pentagons (ASPs). We then show the equinumeracy of these ASPs with Magog-pentagons of a certain shape taking into account the statistic $\rho$. Furthermore we deduce a generating function of these ASPs with respect to the statistic $\rho$ in terms of a Pfaffian and give a partial prove of a conjecture by Behrend from 2018.
Alternating Sign Pentagons and Magog Pentagons
25.04.2023 15:15 - 16:15
Organiser:
M. Drmota
Location: