A Fourfold Refined Enumeration of Alternating Sign Trapezoids

TL;DR

This paper extends the enumeration of alternating sign trapezoids by adding new statistics, establishing joint distribution equivalences with plane partitions, and providing a closed-form for their 2-enumeration.

Contribution

It introduces a new pair of statistics on alternating sign trapezoids and column strict shifted plane partitions, expanding the refined enumeration framework.

Findings

Joint distributions of quadruples of statistics coincide.

Added a new statistic related to the number of -1s.

Derived a closed-form expression for the 2-enumeration.

Abstract

Alternating sign trapezoids have recently been introduced as a generalisation of alternating sign triangles. Fischer established a threefold refined enumeration of alternating sign trapezoids and provided three statistics on column strict shifted plane partitions with the same joint distribution. In this paper, we are able to add a new pair of statistics to these results. More precisely, we consider the number of $-1$s on alternating sign trapezoids and introduce a corresponding statistic on column strict shifted plane partitions that has the same distribution. More generally, we show that the joint distributions of the two quadruples of statistics on alternating sign trapezoids and column strict shifted plane partitions, respectively, coincide. In addition, we provide a closed-form expression for the $2$-enumeration of alternating sign trapezoids.