A probabilistic bijection between twenty-vertex configurations with a free west boundary and Gelfand-Tsetlin patterns avoiding three equal entries in a row

Abstract

We study a coincidence between two enumerations governed by the same product formula, reminiscent of the Robbins numbers: the unweighted enumeration of twenty-vertex configurations on quadrangular domains with fixed west boundary, and the weighted enumeration of Gelfand--Tsetlin patterns avoiding three equal entries in a row. This coincidence naturally raises the question of whether there is a combinatorial explanation relating these two enumerations. In this paper, we provide such an explanation by constructing a probabilistic bijection between twenty-vertex configurations on quadrangular domains and Gelfand--Tsetlin patterns avoiding three equal entries in a row. Under this probabilistic bijection, the west boundary of a twenty-vertex configuration corresponds to the bottom row of Gelfand--Tsetlin patterns; in particular, the fixed boundary case corresponds to Gelfand--Tsetlin patterns with bottom row~$(1, 2, \ldots, n)$. Combining this correspondence with an enumeration formula of Fischer and Schreier-Aigner for Gelfand--Tsetlin patterns avoiding three equal entries in a row with bounded entries, we obtain an enumeration formula for twenty-vertex configurations with a free west boundary.