[Submitted on 20 May 2026 (v1), last revised 6 Jul 2026 (this version, v2)]

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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.

Submission history

From: Atsuro Yoshida [view email]
[v1] Wed, 20 May 2026 13:24:59 UTC (138 KB)
[v2] Mon, 6 Jul 2026 16:44:13 UTC (138 KB)

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