Enumeration of symmetric Gelfand--Tsetlin patterns by linear algebra

TL;DR

This paper provides a linear algebraic proof for counting symmetric Gelfand--Tsetlin patterns with a fixed bottom row, connecting algebraic methods with combinatorial enumeration.

Contribution

It introduces a novel linear algebraic approach to enumerate symmetric Gelfand--Tsetlin patterns, complementing previous combinatorial proofs.

Findings

Linear algebraic proof of enumeration formula

Connection between Gelfand--Tsetlin patterns and lozenge tilings

Determinant manipulations used in enumeration

Abstract

We shall present a ``linear algebraic'' proof (involving some calculations in the algebra of linear operators on a vector space of polynomials and some manipulations of determinants) of the formula for the enumeration of symmetric Gelfand--Tsetlin patterns with fixed bottom row, which was proved by Tri Lai in the context of enumerating symmetric lozenge tilings of a ``halved'' hexagon with ``dents''.