# Refined Enumeration of Halved Monotone Triangles and Applications to Vertically Symmetric Alternating Sign Trapezoids

## TL;DR

This paper develops a weighted enumeration of halved monotone triangles, generalizing VSASM counts, and derives a generating function for vertically symmetric alternating sign trapezoids using operator formulae.

## Contribution

It introduces a new weighted enumeration framework for halved monotone triangles and connects it to generating functions for symmetric alternating sign trapezoids.

## Key findings

- Weighted enumeration formula for halved monotone triangles
- Generating function for vertically symmetric alternating sign trapezoids
- Use of operator formulae in proofs

## Abstract

Halved monotone triangles are a generalisation of vertically symmetric alternating sign matrices (VSASMs). We provide a weighted enumeration of halved monotone triangles with respect to a parameter which generalises the number of $-1$s in a VSASM. Among other things, this enables us to establish a generating function for vertically symmetric alternating sign trapezoids. Our results are mainly presented in terms of constant term expressions. For the proofs, we exploit Fischer's method of operator formulae as a key tool.