Enumeration of symmetric Gelfand--Tsetlin patterns by linear algebra

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''.