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

Summary · qwen2.5:32b

The study establishes a probabilistic bijection between twenty-vertex configurations on quadrangular domains with a free west boundary and Gelfand-Tsetlin patterns that avoid having three consecutive equal entries, revealing a correspondence explained by the alignment of their enumeration formulas. This connection is significant as it links two distinct combinatorial structures through a novel mapping where the west boundary of vertex configurations corresponds to the bottom row of Gelfand-Tsetlin patterns; specifically, when the west boundary is fixed, it aligns with the pattern's bottom row being $(1, 2, \ldots, n)$.

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Exploring the connection between two mathematical enumerations and proving a probabilistic bijection.

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arXiv:2605.21151v2 Announce Type: replace 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.