Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 081, 30 pages      arXiv:2602.04390      https://doi.org/10.3842/SIGMA.2026.081

Colored Interlacing Triangles and Genocchi Medians

Natasha Blitvić a and Leonid Petrov b
a) School of Mathematical Sciences, Queen Mary University of London, UK
b) Department of Mathematics, University of Virginia, Charlottesville, VA, USA

Received February 21, 2026, in final form August 09, 2026; Published online August 25, 2026

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 considered up to a simultaneous relabeling of the colors 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. In particular, under this bijection, the natural decomposition of a triangle into consecutive indecomposable blocks corresponds to the direct-sum decomposition of the associated permutation (readily yielding a generating function identity). Using a recent bijection of Poddar, Sawant, and Shankar (2026), we further identify these `indecomposable' triangles with pure D-permutations. Furthermore, we introduce a $q$-deformation of this enumeration arising naturally from the LLT transition energy. For identity bottom row, we show that the corresponding statistic is the sum of the lower-crossing and lower-nesting statistics on Dumont derangements, and is additive over indecomposable triangles. This yields a new $q$-analogue of the median Genocchi numbers with an elegant continued-fraction expansion, as well as an interpretation of the indecomposable enumerators as Boolean cumulants of a family of probability measures on $\mathbb R$. For arbitrary bottom rows, the failure of additivity is governed by an explicit inversion cross-term, leading to a natural $q$-analogue of the generating function identity for indecomposable triangles. 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, outside of the $N=2$ or $n=3$ regimes.

Key words: colored interlacing triangles; Genocchi medians; Dumont derangements; LLT polynomials.

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