The number of monotone trapezoids with prescribed bottom row

TL;DR

This paper derives a new operator formula for counting monotone trapezoids with a fixed bottom row, generalizing previous results on alternating sign matrices and revealing a hidden operator.

Contribution

It introduces a generalized enumeration formula involving a hidden operator and addresses a novel free boundary condition in the enumeration of monotone trapezoids.

Findings

Established an operator formula for monotone trapezoids with prescribed bottom row.

Discovered a hidden operator that simplifies enumeration in special cases.

Revealed a connection to the coinvariant algebra and proposed a related conjecture.

Abstract

We establish an operator formula for the number of monotone trapezoids with prescribed bottom row, generalizing alternating sign matrices. The special case of the formula for monotone triangles previously provided an alternative proof for the enumeration of alternating sign matrices and led to several results on alternating sign triangles and alternating sign trapezoids. The generalization presented in this paper reveals an additional ``hidden operator'' that is annihilated in the special case of monotone triangles, whose discovery was a major challenge. The enumeration formula is conceptually simple: it applies, in addition to the newly discovered hidden operator, an operator ensuring row strictness to the formula for the number of Gelfand--Tsetlin trapezoids. Notably, the top row of the monotone trapezoid is not prescribed. Thus, our result involves a free boundary, which is a novel situation in this area. We also uncover an unexpected relation to the coinvariant algebra and propose a conjecture on its generalization.