Refined enumerations of alternating sign triangles

TL;DR

This paper refines the enumeration of alternating sign trapezoids using Catalan objects and Motzkin paths, revealing polynomial structures and conjecturing properties of their roots, along with deriving a constant term identity.

Contribution

It introduces a new refinement of alternating sign trapezoids linked to Catalan and Motzkin objects, and explores their polynomial enumeration and root properties.

Findings

Number of trapezoids is polynomial in the shorter base length.

Identified rational roots and formulated conjectures.

Derived a constant term identity for refined counts.

Abstract

This article introduces and investigates a refinement of alternating sign trapezoids by means of Catalan objects and Motzkin paths. Alternating sign trapezoids are a generalisation of alternating sign triangles, which were recently introduced by Ayyer, Behrend and Fischer. We show that the number of alternating sign trapezoids associated with a Catalan object (resp. Motzkin path) is a polynomial function in the length of the shorter base of the trapezoid. We also study the rational roots of the polynomials and formulate several conjectures and derive some partial results. Lastly, we deduce a constant term identity for the refined counting of alternating sign trapezoids.