Alternating Sign Pentagons and Magog Pentagons

TL;DR

This paper introduces alternating sign pentagons, establishes their equinumeracy with Magog pentagons, and derives a Pfaffian generating function considering a key statistic, advancing understanding of these combinatorial objects.

Contribution

It defines alternating sign pentagons, proves their enumeration matches Magog pentagons, and provides a Pfaffian formula for their generating function, extending the theory of alternating sign objects.

Findings

Proved equinumeracy between alternating sign pentagons and Magog pentagons.

Derived a Pfaffian expression for the generating function with respect to a key statistic.

Connected new objects to existing conjectures in combinatorics.

Abstract

Alternating sign triangles have been introduced by Ayyer, Behrend and Fischer in 2016 and it was proven that there is the same number of alternating sign triangles with $n$ rows as there is of $n\times n$ alternating sign matrices. Later on Fischer gave a refined enumeration of alternating sign triangles with respect to a statistic $\rho$, having the same distribution as the unique 1 in the top row of an alternating sign matrix, by connecting alternating sign triangles to $(0,n,n)$- Magog trapezoids. We introduce two more statistics counting the all $0$-columns on the left and right in an alternating sign triangle yielding objects we call alternating sign pentagons. We then show the equinumeracy of these alternating sign pentagons with Magog pentagons of a certain shape taking into account the statistic $\rho$. Furthermore we deduce a generating function of these alternating sign pentagons with respect to the statistic $\rho$ in terms of a Pfaffian and consider the implications of our new results on some open conjectures.