Colored interlacing triangles and Genocchi medians

TL;DR

This paper explores the combinatorial structure of colored interlacing triangles, connecting them to Genocchi medians and Dumont derangements, and introduces a new q-analog related to LLT polynomials.

Contribution

It establishes a bijection between colored interlacing triangles at fixed depth 2 and Dumont derangements, linking to Genocchi medians, and introduces a q-deformation of their enumeration.

Findings

Colored interlacing triangles are in bijection with Dumont derangements at fixed depth 2.

The enumeration of these objects is connected to Genocchi medians.

A new q-analog of the enumeration is introduced, arising from LLT transition energy.

Abstract

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.