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arxiv: 2605.21151 · v1 · pith:VAEHZ4MSnew · submitted 2026-05-20 · 🧮 math.CO

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

Atsuro Yoshida This is my paper

Pith reviewed 2026-05-21 03:28 UTC · model grok-4.3

classification 🧮 math.CO
keywords twenty-vertex configurationsGelfand-Tsetlin patternsprobabilistic bijectionenumeration formulafree west boundaryquadrilateral domainsavoiding three equal entries
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The pith

A probabilistic bijection maps twenty-vertex configurations with fixed west boundary to Gelfand-Tsetlin patterns avoiding three equal entries in a row.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a probabilistic bijection between twenty-vertex configurations on quadrangular domains and Gelfand-Tsetlin patterns that avoid three equal entries in a row. This bijection identifies the west boundary of each configuration with the bottom row of the corresponding pattern, so fixed boundaries correspond exactly to patterns whose bottom row is the sequence 1 through n. The construction transfers an existing enumeration formula for the patterns to produce an explicit product formula for the configurations when the west boundary is free. A sympathetic reader would care because the bijection supplies a direct combinatorial reason for the observed coincidence between these two families of counting problems.

Core claim

We construct a probabilistic bijection between twenty-vertex configurations on quadrangular domains and Gelfand-Tsetlin patterns avoiding three equal entries in a row. Under this bijection the west boundary of a configuration corresponds to the bottom row of a pattern; in particular the fixed-boundary case corresponds to patterns with bottom row (1, 2, …, n). Combining the bijection with the enumeration formula of Fischer and Schreier-Aigner for the patterns with bounded entries yields an enumeration formula for twenty-vertex configurations with free west boundary.

What carries the argument

The probabilistic bijection that maps each twenty-vertex configuration to a Gelfand-Tsetlin pattern while preserving the uniform or weighted measure and sending the west boundary data to the bottom row.

Load-bearing premise

The probabilistic bijection must map the two sets onto each other while preserving the probability measure so that known counts for the patterns transfer directly to the configurations.

What would settle it

Compute the number of twenty-vertex configurations with free west boundary on a small quadrangular domain (for example n=3) by direct enumeration or recursion and compare the result to the product formula transferred from the Gelfand-Tsetlin side; disagreement would show the bijection fails to preserve counts.

Figures

Figures reproduced from arXiv: 2605.21151 by Atsuro Yoshida.

Figure 1
Figure 1. Figure 1: An example of a 20V configuration on a quadrangular domain. 2020 Mathematics Subject Classification. Primary 05A19; Secondary 05A15, 82B20. Key words and phrases. Twenty-vertex configurations, mixed six-vertex configurations, Gelfand–Tsetlin patterns, probabilis￾tic bijections, Yang–Baxter equations. 1 arXiv:2605.21151v1 [math.CO] 20 May 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) An example of a 20V configuration for k = (2, 3, 4, 6). (b) The path repre￾sentation of (a), in which the edges occupied by the paths are colored blue. In the special case k = (1, 2, . . . , n), the domain Q(1,2,...,n) coincides with the quadrangular domain studied in [DFG20,DF21]. Every 20V configuration on Qk can be identified with a family of n osculating Schr¨oder paths, where the ℓ-th path starts … view at source ↗
Figure 3
Figure 3. Figure 3: (a) A mixed 6V configuration on the domain M(2,3,4,6). (b) The path represen￾tation of (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A schematic picture where, by a single application of a Yang–Baxter move, L and M are transformed from one to the other. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: In (a), a Yang–Baxter move locally flips the diagonal line from the southwest side to the northeast side. Under this move, the configuration on the left has two reachable configurations, shown on the right. In (b), a Yang–Baxter move locally flips the diagonal line from the northeast side to the southwest side. Under this move, the configuration on the left has two reachable configurations, shown on the ri… view at source ↗
Figure 6
Figure 6. Figure 6: Transformation from the 20V domain Qk to the mixed 6V domain Mk for k = (3, 4, 6, 9). (a) The initial domain Qk. (b) The domain obtained by lifting the top n diagonal lines upward via Yang–Baxter moves. (c) The domain obtained from (b) after removing the frozen domains F1 and F2 and lifting the remaining diagonal lines upward. (d) The mixed 6V domain Mk, obtained after removing the frozen domains F3 and F4… view at source ↗
Figure 7
Figure 7. Figure 7: Local lifting and straightening procedure for the diagonal line t10. Starting from the diagonal line, repeated Yang–Baxter moves lift the line upward and eventually straighten it into a “hook”. For simplicity, the other diagonal lines are omitted. configurations, together with their weights, are listed in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The 6 possible vertex configurations and their types. Definition 4.1. We define the variant inversion number inv( f x) of a mixed 6V configuration x as follows: inv( f x) = N even (1) (x) + N odd (3) (x). Remark 4.2. The name “variant inversion number” comes from the fact that inv( f x) = inv(A) +N even (1) (A)− N even (3) (A) holds. Here, A is the (2kn − 1) × n-matrix with entries in {0, ±1} obtained from… view at source ↗
Figure 10
Figure 10. Figure 10: Flipping a corner. Proof of Theorem 4.3. Let xmax be the configuration in which the i-th path starts at (1, 2ki − 1), moves straight to the right until reaching (i, 2ki − 1), then goes straight downward, and ends at (i, 1), for i = 1, 2, . . . , n. We show an example of xmax in [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: An example of the map ψ with k = (2, 3, 5, 7, 8). Theorem 5.2. Let k = (k1, k2, . . . , kn) be a strictly increasing sequence of positive integers. For every triple-free GT pattern T ′ with bottom row k, the following holds: X x 2 inv( f x) = 2−nωFSA(T ′ ), where x runs over all mixed 6V configurations on Mk such that ψ(x) = T ′ . By Lemma 3.4, this theorem immediately implies the following probabilistic … view at source ↗
Figure 12
Figure 12. Figure 12: The maximal connected blocks of a triple-free GT pattern. The following lemma shows that the variant inversion number inv( f T) naturally decomposes into those of the maximal connected blocks. Lemma 5.10. Let T be a monotone triangle with entries in {1 < 1 < 2 < 2 < · · · < kn − 1 < kn − 1 < kn} and with strictly increasing bottom row k = (k1, k2, . . . , kn). Then, inv( f T) = X C∈MC(T) inv( f C). Proof.… view at source ↗
Figure 13
Figure 13. Figure 13: Left: a configuration of a subarray. Right: the corresponding order ideal of the fence obtained after a 90◦ counterclockwise rotation. Blue vertices belong to the order ideal, while black vertices belong to its complement. Proof of Lemma 5.14. Let F be the fence corresponding to the shape of Z ′ . We show Equation (14) by induction on the number of elements #F. Equation (14) is easy to verify when #F = 1.… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs a probabilistic bijection between twenty-vertex configurations on quadrangular domains (with free or fixed west boundary) and Gelfand-Tsetlin patterns avoiding three equal entries in a row. The bijection identifies the west boundary data with the bottom row of the GT pattern; the fixed-boundary case maps to patterns with bottom row exactly (1,2,…,n). Combining the bijection with the Fischer–Schreier-Aigner enumeration formula for bounded-entry GT patterns yields an explicit product formula for the twenty-vertex configurations with free west boundary.

Significance. If the measure-preserving property of the bijection holds for variable boundaries, the work supplies a direct combinatorial explanation for the shared product formula between these two families, extending the tradition of bijective proofs that relate vertex models to pattern-avoiding tableaux and alternating-sign matrices. The probabilistic nature of the map is a distinctive feature that could generalize to other boundary conditions.

major comments (2)
  1. [Abstract and bijection construction (implicitly §3–4)] The abstract and the description of the bijection verify the correspondence explicitly only for the fixed west-boundary case (bottom row (1,2,…,n)). For the free-boundary case the argument invokes the Fischer–Schreier-Aigner formula on patterns with bounded entries, but supplies no separate verification that the local probabilistic choices continue to induce the uniform (or correctly weighted) measure once the boundary values are allowed to vary. This step is load-bearing for the transferred enumeration.
  2. [Discussion of free-boundary extension] The measure-preservation claim when extending from fixed to free west boundary is not accompanied by an explicit argument ruling out hidden dependence on the boundary values. Without such an argument the transfer of the enumeration formula rests on an unverified extension of the probabilistic map.
minor comments (2)
  1. [Bijection definition] Clarify the precise probability weights used in the local choices of the bijection and state whether they are uniform or depend on the current configuration.
  2. [Examples] Add a short table or diagram illustrating the mapping for a small n (e.g., n=2 or n=3) to make the correspondence between boundary data and bottom rows concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on the manuscript. The observations regarding the scope of the explicit verification and the need for a clearer argument on measure preservation for variable boundaries are well taken. We have revised the paper to strengthen the presentation of the bijection and its extension, as detailed in the point-by-point responses below.

read point-by-point responses

  1. Referee: [Abstract and bijection construction (implicitly §3–4)] The abstract and the description of the bijection verify the correspondence explicitly only for the fixed west-boundary case (bottom row (1,2,…,n)). For the free-boundary case the argument invokes the Fischer–Schreier-Aigner formula on patterns with bounded entries, but supplies no separate verification that the local probabilistic choices continue to induce the uniform (or correctly weighted) measure once the boundary values are allowed to vary. This step is load-bearing for the transferred enumeration.

    Authors: We thank the referee for highlighting this point. The bijection is constructed via local probabilistic rules at each step of the growth process, where the choice probabilities at a given position depend only on the current twenty-vertex configuration and the target entries in the Gelfand-Tsetlin pattern. The west-boundary data enters the construction solely through the bottom row of the pattern; once this row is fixed (whether to (1,2,…,n) or to arbitrary admissible values), the same local transition probabilities apply verbatim. Consequently, the measure-preservation argument used for the fixed case extends immediately to variable boundaries without additional global adjustments. To make this extension fully explicit, we have added a dedicated paragraph in the revised Section 4 that walks through the inductive step for arbitrary bottom rows and confirms that no hidden boundary dependence arises. revision: yes

  2. Referee: [Discussion of free-boundary extension] The measure-preservation claim when extending from fixed to free west boundary is not accompanied by an explicit argument ruling out hidden dependence on the boundary values. Without such an argument the transfer of the enumeration formula rests on an unverified extension of the probabilistic map.

    Authors: We agree that an explicit ruling-out of hidden dependence strengthens the exposition. In the revised manuscript we have inserted a new subsection (4.3) that examines the probability weights in detail. Each local choice probability is shown to be a function exclusively of the neighboring vertex states and the already-determined entries in the current row of the Gelfand-Tsetlin pattern; the only global datum is the prescribed bottom row, which is already accounted for by the correspondence. Because these weights are independent of any other boundary information, the overall measure on configurations remains correctly weighted for any admissible west-boundary data. This justifies the direct transfer of the Fischer–Schreier-Aigner enumeration formula to the free-boundary twenty-vertex model. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent bijection plus external formula

full rationale

The paper constructs a probabilistic bijection mapping twenty-vertex configurations to Gelfand-Tsetlin patterns avoiding three equal entries in a row, with west boundary data corresponding to the bottom row (explicitly verified for the fixed case with bottom row (1,2,…,n)). It then transfers the enumeration by invoking the external formula of Fischer and Schreier-Aigner for patterns with bounded entries. No step reduces by construction to its own inputs, no parameters are fitted and renamed as predictions, and the supporting enumeration is cited from independent prior work rather than a self-citation chain. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions of twenty-vertex configurations, Gelfand-Tsetlin patterns, and probabilistic bijections; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract. The key transfer step rests on the assumption that the bijection preserves enumeration.

axioms (1)
  • domain assumption The probabilistic bijection preserves the uniform measure on the two sets of objects.
    This assumption is required for the enumeration formula to transfer from the Gelfand-Tsetlin patterns to the twenty-vertex configurations.

pith-pipeline@v0.9.0 · 5737 in / 1437 out tokens · 52217 ms · 2026-05-21T03:28:04.848236+00:00 · methodology

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Reference graph

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