The operator formula for monotone triangles - simplified proof and three generalizations

TL;DR

This paper offers a simplified proof of an operator formula for counting monotone triangles with a given bottom row, and extends it to three new generalizations, including weighted enumeration related to alternating sign matrices.

Contribution

The paper introduces a simplified proof of the operator formula and presents three novel generalizations, including weighted enumeration of monotone triangles.

Findings

Simplified proof of the operator formula

Three new generalizations of the enumeration formula

Weighted enumeration connecting to alternating sign matrices

Abstract

We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted enumeration of monotone triangles which specializes to the weighted enumeration of alternating sign matrices with respect to the number of -1s in the matrix when prescribing $(1,2,...,n)$ as the bottom row of the monotone triangle.