arXiv subjects

Lauren Engelthaler

Publications and source records attributed to Lauren Engelthaler.

3 recordsLinked to original sources

arXiv2026

Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux

Young tableaux are fundamental objects in algebraic combinatorics and representation theory, with operations such as promotion and jeu de taquin playing a central role in their structure and applications. While these operations are well understood for finite tableaux, their behavior on infinite tableaux has so far been studied mainly within probabilistic frameworks. In this paper, we investigate jeu de taquin on infinite standard Young tableaux from a purely combinatorial and dynamical point of view. We analyze the action of jeu de taquin on infinite shapes, describe the structure of inverse images, and classify tableaux exhibiting periodic, pre-periodic, and recurrent behavior. We also introduce a natural metric on the space of infinite tableaux and show that jeu de taquin defines a chaotic dynamical system in the sense of Devaney. These results extend classical tableau theory to infinite settings and identify connections between combinatorial dynamics and infinite representation-theoretic structures.

math.CO↗

arXiv2024

Integer diagonal forms for subset intersection relations

For integers $0 \leq \ell \leq k_{r} \leq k_{c} \leq n$, we give a description for the Smith group of the incidence matrix with rows (columns) indexed by the size $k_r$ ($k_c$, respectively) subsets of an $n$-element set, where incidence means intersection in a set of size $\ell$. This generalizes work of Wilson and Bier from the 1990s which dealt only with the case where incidence meant inclusion. Our approach also describes the Smith group of any matrix in the $\mathbb{Z}$-linear span of these matrices so includes all integer matrices in the Bose-Mesner algebra of the Johnson association scheme: for example, the association matrices themselves as well as the Laplacian, signless Laplacian, Seidel adjacency matrix, etc. of the associated graphs. In particular, we describe the critical (also known as sandpile) groups of these graphs. The complexity of our formula grows with the parameters $k_{r}$ and $k_{c}$, but is independent of $n$ and $\ell$, which often leads to an efficient algorithm for computing these groups. We illustrate our techniques to give diagonal forms of matrices attached to the Kneser and Johnson graphs for subsets of size $3$, whose invariants have never before been described, and recover results from a variety of papers in the literature in a unified way.

math.CO↗

arXiv2021

A combinatorial approach to counting primitive periodic and primitive pseudo orbits on circulant graphs

For families of 4-regular directed circulant graphs with $n$ vertices, we count the number of primitive periodic orbits of length up to at least $n$. The relevant counting techniques are then extended to count the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) of length up to at least $n$ that lack self-intersections, or that self-intersect only at individual vertices repeated exactly twice (2-encounters of length zero), for two particular families of 4-regular directed circulant graphs. We then regard these two families of graphs as families of quantum graphs and use the counting results to compute the variance of the coefficients of the quantum graph's characteristic polynomial.

math.CO↗