arXiv subjects

William Q. Erickson

Publications and source records attributed to William Q. Erickson.

26 records · Page 2Linked to original sources

arXiv2023

Young tableau reconstruction via minors

The tableau reconstruction problem, posed by Monks (2009), asks the following. Starting with a standard Young tableau $T$, a 1-minor of $T$ is a tableau obtained by first deleting any cell of $T$, and then performing jeu de taquin slides to fill the resulting gap. This can be iterated to arrive at the set of $k$-minors of $T$. The problem is this: given $k$, what are the values of $n$ such that every tableau of size $n$ can be reconstructed from its set of $k$-minors? For $k=1$, the problem was recently solved by Cain and Lehtonen. In this paper, we solve the problem for $k=2$, proving the sharp lower bound $n \geq 8$. In the case of multisets of $k$-minors, we also give a lower bound for arbitrary $k$, as a first step toward a sharp bound in the general multiset case.

math.CO↗

arXiv2023

A Cayley-Menger formula for the earth mover's simplex

The earth mover's distance (EMD) is a well-known metric on spaces of histograms; roughly speaking, the EMD measures the minimum amount of work required to equalize two histograms. The EMD has a natural generalization that compares an arbitrary number of histograms; in this case, the EMD can be viewed as hypervolume in $d$ dimensions, where the histograms are vertices of a $d$-simplex. For $d=2$, it is known that the EMD between three histograms equals half the sum of the pairwise EMDs -- a sort of Heron's formula for histograms, but where the area equals the semiperimeter. In this paper, by introducing an object we call the earth mover's simplex, we prove two generalizations of this Heron-like formula in arbitrary dimension: the first (a sort of Cayley-Menger formula) expresses the EMD in terms of the edge lengths (the pairwise EMDs), the second in terms of the facets (EMDs excluding one histogram).

math.CO↗

arXiv2023

The sum of all width-one tensors

This paper generalizes a recent result by the authors concerning the sum of width-one matrices; in the present work, we consider width-one tensors of arbitrary dimensions. A tensor is said to have width 1 if, when visualized as an array, its nonzero entries lie along a path consisting of steps in the directions of the standard coordinate vectors. We prove two different formulas to compute the sum of all width-one tensors with fixed dimensions and fixed sum of (nonnegative integer) components. The first formula is obtained by converting width-one tensors into tuples of one-row semistandard Young tableaux; the second formula, which extracts coefficients from products of multiset Eulerian polynomials, is derived via Stanley-Reisner theory, making use of the EL-shelling of the order complex on the standard basis of tensors.

math.CO↗

arXiv2023

Comparing weighted difference and earth mover's distance via Young diagrams

We consider two natural statistics on pairs of histograms, in which the $n$ bins have weights $0, \ldots, n-1$. The difference ($\mathbf{D}$) between the weighted totals of the histograms is, in a sense, refined by the earth mover's distance ($\mathbf{EMD}$), which measures the amount of work required to equalize the histograms. We were recently surprised, however, by how little $\mathbf{EMD}$ actually does refine $\mathbf{D}$ in certain real-world applications, which led to the main problem in this paper: what is the probability that $\mathbf{EMD} = |\mathbf{D}|$? We derive a formula for this probability, as well as the expected value of $|\mathbf{D}|$, via the combinatorics of Young diagrams and plane partitions. We then generalize our results to an arbitrary number of histograms, where we realize this higher-dimensional $\mathbf{D}$ as distance on the Type-A root lattice.

math.CO↗

arXiv2022

Klein cordial trees and odd cyclic cordial friendship graphs

For a graph $G$ and an abelian group $A$, a labeling of the vertices of $G$ induces a labeling of the edges via the sum of adjacent vertex labels. Hovey introduced the notion of an $A$-cordial vertex labeling when both the vertex and edge labels are as evenly distributed as possible. Much work has since been done with trees, hypertrees, paths, cycles, ladders, prisms, hypercubes, and bipartite graphs. In this paper we show that all trees are $\mathbb{Z}_2^2$-cordial except for $P_4$ and $P_5$. In addition, we give numerous results relating to $\mathbb{Z}_m$-cordiality of the friendship graph $F_n$. The most general result shows that when $m$ is an odd multiple of $3$, then $F_n$ is $\mathbb{Z}_m$-cordial for all $n$. We also give a general conjecture to determine when $F_n$ is $\mathbb{Z}_m$-cordial.

math.CO↗

arXiv2021

Enright resolutions encoded by a generating function for Blattner's formula: Type A

Consider the classical action of ${\rm GL}_n$ on a sum of $q$ copies of the defining representation and $p$ copies of its dual; by Howe duality, the polynomial functions on this space decompose under the joint action of ${\rm GL}_n$ and $\mathfrak{gl}_{p+q}$. The modules for $\mathfrak{gl}_{p+q}$ are infinite-dimensional and their structure is complicated outside a certain stable range, although Enright and Willenbring (2005) constructed resolutions in terms of generalized Verma modules. We show that these resolutions can be read off from the coefficients in a formal series arising in an entirely different setting: discrete series representations of ${\rm SU}(n,p+q)$ in the case of two noncompact simple roots.

math.RT↗

arXiv2020

Contingency tables and the generalized Littlewood-Richardson coefficients

The Littlewood-Richardson coefficients $c^λ_{μν}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^λ_n$ in the tensor product of polynomial representations $F^μ_n\otimes F^ν_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.

math.RT↗

arXiv2020

A generalization for the expected value of the earth mover's distance

The earth mover's distance (EMD), also called the first Wasserstein distance, can be naturally extended to compare arbitrarily many probability distributions, rather than only two, on the set $[n]=\{1,\dots,n\}$. We present the details for this generalization, along with a highly efficient algorithm inspired by combinatorics; it turns out that in the special case of three distributions, the EMD is half the sum of the pairwise EMD's. Extending the methods of Bourn and Willenbring (arXiv:1903.03673), we compute the expected value of this generalized EMD on random $d$-tuples of distributions, using a generating function which coincides with the Hilbert series of the Segre embedding. We then use the EMD to analyze a real-world data set of grade distributions.

math.CO↗