Colored interlacing triangles and Genocchi medians

TL;DR

This work resolves the depth-$2$ enumeration of colored interlacing triangles by establishing a bijection with Dumont derangements, yielding $T_2(n)=n!\,H_n$ where $H_n$ are Genocchi medians, and introduces a natural $q$-deformation $T_2(n;q)$ arising from the LLT transition energy, producing new $q$-analogs of Genocchi medians. It also defines the inter-level statistic $\psi$ to weight $N$-level transitions and proves that for $N=2$ the corresponding polynomials $T_2(n;q)$ are palindromic with degree $\binom{n}{2}$ and divisibility by $2^{n-1}$, while revealing structural properties such as linear cumulants and refined bottom-row decompositions. The paper provides substantial computational data, including GPU-accelerated enumeration of $T_N(n)$ and explicit $P_n(q)$ for $n\le 9$, and develops a Markov chain Monte Carlo sampler for the $q$-weighted depth-$2$ ensemble, revealing a Mallows-like permutation structure in the limit. Together, these results connect the LLT-based probabilistic model to classical combinatorics (Genocchi medians, Dumont derangements) and delineate the computational and combinatorial boundaries of tractability in the $(N,n)$ parameter space.

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.

Figures (4)

• Figure 1: Left: A colored interlacing triangle with $n=3$ colors and $N=3$ levels. Colors: 1 (red), 2 (green), 3 (blue). Right: indexing notation $\lambda[i]^k_j$ for the $i$-th triangle. Row sequences: $\lambda^{1}=(\textcolor{green!70!black}{2},\textcolor{blue!70}{3},\textcolor{red!80}{1})$, $\lambda^{2}=(\textcolor{green!70!black}{2},\textcolor{red!80}{1},\textcolor{blue!70}{3},\textcolor{blue!70}{3},\textcolor{green!70!black}{2},\textcolor{red!80}{1})$, $\lambda^{3}=(\textcolor{green!70!black}{2},\textcolor{red!80}{1},\textcolor{green!70!black}{2},\textcolor{blue!70}{3},\textcolor{blue!70}{3},\textcolor{red!80}{1},\textcolor{blue!70}{3},\textcolor{green!70!black}{2},\textcolor{red!80}{1})$. Each pair of consecutive rows may be written in one line using the linear order \ref{['eq:interlacing_order']}, for example, $(2|2|13|3|32|1|1)$ for the first two levels.
• Figure 2: Sampled depth-$2$ colored interlacing triangles with $n=50$ colors, ordered as a rainbow from color $1$ (red) to color $50$ (violet). The two triangles were obtained after MCMC runs of $10^7$ steps each. Top: $q=0.2$ (here the $\psi$-statistic is $55$). Bottom: $q=0.98$ (here $\psi=541$). Lines connect each color's occurrences across levels, and indicate the interlacing properties.
• Figure 3: Position frequency heatmaps from $10^5$ samples collected after $10^6$ "burn-in" steps, with $10^4$ steps between samples (so, a total of $10^9$ MCMC steps), where $n=25$. Left column: Level $1$ ($n \times n$). Right column: Level $2$ ($n \times 2n$). Top row: $q=0.2$. Bottom row: $q=0.9$.
• Figure 4: Empirical distribution of $\psi$ with $n=25$, $q=0.9$, from the same samples as in \ref{['fig:sampling_heatmaps']}.

Theorems & Definitions (33)

• Definition 1.1
• Definition 2.1
• Theorem 2.2
• proof
• Corollary 2.3
• proof
• Proposition 2.4
• proof
• Remark 2.5
• Definition 3.1: ABW2023coloured_LLT