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

TL;DR

This paper establishes a probabilistic bijection linking twenty-vertex configurations with fixed boundary conditions to Gelfand-Tsetlin patterns avoiding three equal entries, explaining a shared enumeration formula.

Contribution

It introduces a probabilistic bijection that explains the combinatorial coincidence between two seemingly unrelated enumeration problems.

Findings

Established a probabilistic bijection between twenty-vertex configurations and Gelfand-Tsetlin patterns.

Derived an enumeration formula for twenty-vertex configurations with a free west boundary.

Connected boundary conditions of configurations to specific bottom rows in Gelfand-Tsetlin patterns.

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 pattern avoiding three equal entries in a row with bounded entries, we obtain an enumeration formula for twenty-vertex configurations with a free west boundary.