Colored interlacing triangles and Genocchi medians

Colored interlacing triangles, introduced by Aggarwal-Borodin-Wheeler (2024), provide the combinatorial framework for the Central Limit Theorem for probability measures arising from the Lascoux-Leclerc-Thibon (LLT) polynomials. Colored interlacing triangles depend on two key parameters: the number of colors $n$ and the depth of the triangle $N$. Recent work of Gaetz-Gao (2025) connects these objects to Schubert calculus and resolves the enumeration for $n=3$ and arbitrary depth $N$. However, the enumerative behavior for general $n$ has remained open. In this paper, we analyze the complementary regime: fixed depth $N=2$ and arbitrary number of colors $n$. We prove that in this setting, colored interlacing triangles are in bijection with Dumont derangements, identifying their enumeration with the Genocchi medians. This connects the probabilistic model to a rich hierarchy of classical combinatorial objects. Furthermore, we introduce a $q$-deformation of this enumeration arising naturally from the LLT transition energy. This yields new $q$-analogs of the Genocchi medians. Finally, we present computational results and sampling algorithms for colored interlacing triangles with higher $N$ or $n$, which suggests the limits of combinatorial tractability in the $(N,n)$ parameter space.

💡 Research Summary

This paper presents a comprehensive study of “colored interlacing triangles,” combinatorial objects introduced by Aggarwal-Borodin-Wheeler (2024) as the discrete component in the Central Limit Theorem for probability measures arising from LLT (Lascoux-Leclerc-Thibon) polynomials. These structures are parameterized by the number of colors n and the depth N, forming a vast combinatorial landscape.

The authors address the enumerative challenge by focusing on the complementary regime where the depth is fixed at N=2 and the number of colors n is arbitrary. Their primary achievement is a bijective proof demonstrating that colored interlacing triangles of depth 2 are in one-to-one correspondence with “Dumont derangements,” a classical class of permutations. Consequently, the number of such triangles (up to a factor of n! for color permutations) is exactly the n-th Genocchi median H_n. This result firmly connects the probabilistic model underlying LLT polynomials to a well-established hierarchy of combinatorial numbers.

Building on this enumeration, the paper explores a natural q-deformation of the problem stemming from the “inter-level transition energy” present in the LLT Central Limit Theorem. Applying this q-weight, defined via a statistic ψ(λ_{k-1}, λ_k), to the depth-2 triangles yields a new polynomial q-analog of the Genocchi medians. The authors show that this new q-analog is distinct from previously known ones (e.g., those based on the inversion number, the Denert statistic, or the Seidel triangle recurrence), thus enriching the family of q-Genocchi numbers.

Finally, the paper investigates the limits of tractability in the broader (N, n) parameter space. Through computational enumeration for small parameters, the authors provide data on T_N(n), which refutes a specific conjecture from prior literature regarding the case n=4. The observed rapid growth and presence of large prime factors suggest that simple closed-form formulas may not exist for general N and n. To probe this complexity, the authors also develop a Markov Chain Monte Carlo sampling algorithm for the q-weighted distribution on depth-2 triangles, suggesting a path for algorithmic exploration in less tractable regions.

In summary, this work resolves the enumeration for the horizontal slice (N=2) of the colored interlacing triangle hierarchy, linking it to Genocchi medians, introduces a new significant q-analog from probabilistic considerations, and computationally maps the challenging terrain of the general problem, highlighting the intersection of probability, algebraic combinatorics, and computational enumeration.