Refined enumerations of alternating sign matrices: monotone (d,m)-trapezoids with prescribed top and bottom row

TL;DR

This paper introduces monotone (d,m)-trapezoids, generalizes enumeration formulas for alternating sign matrices, and connects these objects to polynomial expansions, providing explicit formulas and settling a recent conjecture.

Contribution

It establishes that the count of monotone (d,m)-trapezoids with fixed rows appears as a coefficient in polynomial expansions, generalizing previous conjectures and linking to alternating sign matrix enumeration.

Findings

Derived explicit formulas for monotone trapezoids counts.

Connected trapezoid counts to polynomial coefficients in matrix enumeration.

Extended enumeration techniques to partial alternating sign matrices.

Abstract

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to $n \times n$ alternating sign matrices when prescribing $(1,2,...,n)$ as bottom row of the array. We define monotone $(d,m)$--trapezoids as monotone triangles with $m$ rows where the $d-1$ top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row $(k_1,...,k_n)$ is given by a polynomial $\alpha(n;k_1,...,k_n)$ in the $k_i$'s. The main purpose of this paper is to show that the number of monotone $(d,m)$--trapezoids with prescribed top and bottom row appears as a coefficient in the expansion of a specialization of $\alpha(n;k_1,...,k_n)$ with respect to a certain polynomial basis. This settles a generalization of a recent conjecture of Romik and the author. Among other things, the result is used to express the number of monotone triangles with bottom row $(1,2,...,i-1,i+1,...,j-1,j+1,...,n)$ (which is, by the standard bijection, also the number of $n \times n$ alternating sign matrices with given top two rows) in terms of the number of $n \times n$ alternating sign matrices with prescribed top and bottom row, and, by a formula of Stroganov for the latter numbers, to provide an explicit formula for the first numbers. (A formula of this type was first derived by Karklinsky and Romik using the relation of alternating sign matrices to the six--vertex model.)