arXiv subjects

William Q. Erickson

Publications and source records attributed to William Q. Erickson.

At least 19 recordsLinked to original sources

arXiv2026

A lattice path model for the volume of the Monge polytope

Monge matrices arise throughout combinatorial optimization and algorithm design; the Monge polytope $\mathcal{M}_{pq}$ is the set of $p \times q$ Monge matrices lying inside the standard simplex on the set of matrix coordinates. We find a Stanley decomposition of the associated affine semigroup, and use it to obtain a volume formula for $\mathcal{M}_{pq}$ expressed as a sum over "Z-avoiding" Delannoy paths in a $p \times q$ grid. An efficient dynamic-programming implementation of this formula computes the volume in dimensions far beyond the reach of general-purpose exact-volume algorithms (e.g., the volume of $\mathcal{M}_{20,20}$, which has dimension 399, is computed in a fraction of a second). As a corollary of our Stanley decomposition, we also obtain a combinatorial closed form for the Ehrhart series of $\mathcal{M}_{pq}$.

math.CO↗

arXiv2026

Homometric subsets of $\mathbb{Z}_n$ with cardinality 5: classification and enumeration

Two subsets of $\mathbb{Z}_n$ are said to be homometric if they have the same multiset of pairwise cyclic (i.e., Lee) distances. Homometric subsets necessarily have the same cardinality, say $k$. In this paper, for all positive integers $n$, we classify the homometric subsets of $\mathbb{Z}_n$ with cardinality $k=5$ (modulo cyclic shifts and reflections). Our classification consists of six families of homometric pairs, and one family of homometric triples. We also give a closed-form generating function that counts these homometric pairs and triples for all $n$. The same problem for $k \leq 4$ was partially solved by Erdős and ultimately settled by Rosenblatt-Berman (1984). As an immediate application of our result, one obtains an explicit criterion for the solvability of the crystallographic phase retrieval problem, in the setting of binary signals supported on $k=5$ many atoms.

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arXiv2026

Explicit marginal distributions for permutations with prescribed Robinson-Schensted shape

Given a permutation $σ$, the Robinson-Schensted correspondence determines a certain partition called the shape of $σ$. Famously, the shape measures the longest unions of increasing and decreasing subsequences, thus giving global information about $σ$. In this paper, by contrast, we ask how prescribing a shape collectively controls local behavior: namely, if $σ$ is a random permutation of shape $λ$, then what is $P^λ_{ij} :=$ the probability that $σ(i) = j$? Using tableau-theoretic methods, we derive explicit formulas for $P^λ_{ij}$ when $λ$ is a hook, two-row, or rectangular shape. We use these formulas to depict and analyze the intricate diffraction-like patterns in the matrices $(P^λ_{ij})$. As a surprising application, we show that for both hook and two-row shapes, as the largest part of $λ$ tends to infinity with the remaining parts fixed (summing to $m$), the expected proportion of fixed points in $σ$ approaches the Wallis integral $\int_0^{π/2} \sin^{2m+1} x \: dx = (2m)!! / (2m+1)!!$.

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arXiv2026

A combinatorial interpretation of the Bernstein degree of unitary highest weight modules

The Bernstein degree ($\operatorname{Deg}$) is a fundamental invariant of admissible representations of a real reductive Lie group $G_{\mathbb{R}}$. Our main result concerns the classical dual pairs $(G_{\mathbb{R}}, H_{\mathbb{R}}(k))$, namely $(\operatorname{U}(p,q), \: \operatorname{U}(k))$, $(\operatorname{Mp}(2n, \mathbb{R}), \: \operatorname{O}(k))$, and $(\operatorname{O}^*(2n), \: \operatorname{Sp}(k))$, where $k$ is any positive integer. In this setting, via Howe duality, each irreducible representation $σ$ of $H_{\mathbb{R}}(k)$ corresponds to a unitary highest weight module $L_{λ(σ)}$ for $G_{\mathbb{R}}$. A landmark result of Nishiyama-Ochiai-Taniguchi (2001) expressed $\operatorname{Deg} L_{λ(σ)}$ as a product of two quantities: the dimension of $σ$ and the degree of the associated variety. However, this result was limited to a specific range of the parameter $k$ (namely $k \leq r$, the real rank of $G_{\mathbb{R}}$). The present paper resolves this limitation by introducing, for all $k$, the combinatorial interpretation $\operatorname{Deg} L_{λ(σ)} = \#( \mathcal{Q}_k(σ) \times \mathcal{P}_k)$, where $\mathcal{Q}_k(σ)$ is a certain set of semistandard tableaux and $\mathcal{P}_k$ is a set of plane partitions. (The result remains partly conjectural in the $\operatorname{Mp}(2n, \mathbb{R})$ case.) Beyond the dual pair setting, we generalize the set $\mathcal{P}_k$ to all groups $G_{\mathbb{R}}$ of Hermitian type, and we exhibit analogues of the Nishiyama-Ochiai-Taniguchi result for certain families of unitary highest weight modules of $\operatorname{E}_6$ and $\operatorname{E}_7$.

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arXiv2026

Stanley decompositions of modules of covariants

Let $H$ be a complex reductive group, with finite-dimensional representations $W$ and $U$. The module of covariants for $W$ of type $U$ is the space of all $H$-equivariant polynomial maps $φ: W \longrightarrow U$. In this paper, we take $H$ to be one of the classical groups $\operatorname{GL}(V)$, $\operatorname{O}(V)$, or $\operatorname{Sp}(V)$, where $W$ is a direct sum of copies of $V$ and $V^*$, and $U$ is an arbitrary rational representation (with $U$ restricted to exterior powers of $V$ in the $H= \operatorname{O}(V)$ case). Our main result gives uniform Stanley decompositions of these modules of covariants, with Stanley spaces parametrized by combinatorial objects we call jellyfish. As a corollary, we write down the Hilbert series as a finite sum of rational functions, each with a combinatorial interpretation in terms of lattice paths. Notably, these results do not rely on the module being Cohen-Macaulay. We further apply our methods to invariant rings for $\operatorname{SL}(V)$ and $\operatorname{SO}(V)$. Our proofs (for $H = \operatorname{GL}(V)$ and $\operatorname{Sp}(V)$) rely on previous work by Jackson on standard monomial theory for dual reductive pairs, since classical modules of covariants can be viewed via Howe duality as Harish-Chandra modules of unitary highest weight representations of a certain real reductive group. As a first step toward extending this program to arbitrary unitary highest weight representations (including those of the exceptional groups), we establish analogous results uniformly for the Wallach representations of type ADE.

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arXiv2025

Unitarity of highest weight Harish-Chandra modules and smoothness of Schubert varieties

Let $G_{\mathbb{R}}$ be a Lie group of Hermitian type, and $L(λ)$ a highest weight Harish-Chandra module of $G_{\mathbb{R}}$ with highest weight $λ$. In this article, we exhibit a bijection between the set of connected Dynkin subdiagrams containing the noncompact simple root and the set of unitary highest weight modules $L(-wρ-ρ)$, where $ρ$ is half the sum of positive roots. We find that $L(-wρ-ρ)$ is unitary if and only if the Schubert variety $X(w)$ is smooth. We also give the cardinality of the set of unitary highest weight modules $L(-wρ-ρ)$ for each Kazhdan-Lusztig right cell.

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arXiv2025

Palindromicity of the numerator of a statistical generating function

We prove a conjecture of Bourn and Willenbring (2020) regarding the palindromicity and unimodality of a certain family of polynomials $N_n(t)$. These recursively defined polynomials arise as the numerators of generating functions in the context of the discrete one-dimensional earth mover's distance (EMD). The key to our proof is showing that the defining recursion can be viewed as describing sums of symmetric differences of pairs of Young diagrams; in this setting, palindromicity is equivalent to the preservation of the symmetric difference under the transposition of diagrams. We also observe a connection to recent work by Defant et al. (2024) on the Wiener index of minuscule lattices, which we reinterpret combinatorially to obtain explicit formulas for the coefficients of $N_n(t)$ and for the expected value of the discrete EMD.

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arXiv2025

Dimension identities, almost self-conjugate partitions, and BGG complexes for Hermitian symmetric pairs

An almost self-conjugate (ASC) partition has a Young diagram in which each arm along the diagonal is exactly one box longer than its corresponding leg. Classically, the ASC partitions and their conjugates appear in two of Littlewood's symmetric function identities. These identities can be viewed as Euler characteristics of BGG complexes of the trivial representation, for classical Hermitian symmetric pairs. In this paper, we consider partitions in which the arm-leg difference is an arbitrary constant $m$. By viewing these partitions as highest weights, we establish an infinite family of dimension identities between $\mathfrak{gl}_n$- and $\mathfrak{gl}_{n+m}$-modules. We then interpret this result in the context of blocks in parabolic category $\mathcal{O}$: in particular, we exhibit six infinite families of congruent blocks whose corresponding posets of highest weights consist of the partitions in question. These posets, in turn, lead to generalizations of the Littlewood identities and their corresponding BGG complexes. Our results in this paper shed light on the surprising combinatorics underlying the work of Enright and Willenbring (2004).

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arXiv2025

$K$-type multiplicities in degenerate principal series via Howe duality

Let $K$ be one of the complex classical groups ${\rm O}_k$, ${\rm GL}_k$, or ${\rm Sp}_{2k}$. Let $M \subseteq K$ be the block diagonal embedding ${\rm O}_{k_1} \times \cdots \times {\rm O}_{k_r}$ or ${\rm GL}_{k_1} \times \cdots \times {\rm GL}_{k_r}$ or ${\rm Sp}_{2k_1} \times \cdots \times {\rm Sp}_{2k_r}$, respectively. By using Howe duality and seesaw reciprocity as a unified conceptual framework, we prove a formula for the branching multiplicities from $K$ to $M$ which is expressed as a sum of generalized Littlewood-Richardson coefficients, valid within a certain stable range. By viewing $K$ as the complexification of the maximal compact subgroup $K_{\mathbb{R}}$ of the real group $G_{\mathbb{R}} = {\rm GL}(k,\mathbb{R})$, ${\rm GL}(k, \mathbb{C})$, or ${\rm GL}(k,\mathbb{H})$, respectively, one can interpret our branching multiplicities as $K_{\mathbb{R}}$-type multiplicities in degenerate principal series representations of $G_{\mathbb{R}}$. Upon specializing to the minimal $M$, where $k_1 = \cdots = k_r = 1$, we establish a fully general tableau-theoretic interpretation of the branching multiplicities, corresponding to the $K_{\mathbb{R}}$-type multiplicities in the principal series.

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arXiv2025

Tensor invariants for classical groups revisited

We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\otimes p} \otimes V^{*\otimes q}$, where $G$ is one of the classical groups ${\rm GL}(V)$, ${\rm SL}(V)$, ${\rm O}(V)$, ${\rm SO}(V)$, or ${\rm Sp}(V)$. Spanning sets for the invariant subspace have long been well known, but linear bases are more delicate. The main contribution of this paper is a combinatorial realization of linear bases via standard Young tableaux and arc diagrams, in a uniform manner for all five classical groups. As a secondary contribution, we survey the many equivalent ways -- some old, some new -- to enumerate the elements in these bases.

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arXiv2024

Robinson-Schensted shapes arising from cycle decompositions

In the symmetric group $S_n$, each element $σ$ has an associated cycle type $α$, a partition of $n$ that identifies the conjugacy class of $σ$. The Robinson-Schensted (RS) correspondence links each $σ$ to another partition $λ$ of $n$, representing the shape of the pair of Young tableaux produced by applying the RS row-insertion algorithm to $σ$. Surprisingly, the relationship between these two partitions, namely the cycle type $α$ and the RS shape $λ$, has only recently become a subject of study. In this work, we explicitly describe the set of RS shapes $λ$ that can arise from elements of each cycle type $α$ in cases where $α$ consists of two cycles. To do this, we introduce the notion of an $α$-coloring, where one colors the entries in a certain tableau of shape $λ$, in such a way as to construct a permutation $σ$ with cycle type $α$ and RS shape $λ$.

math.CO↗

arXiv2024

Expected value and a Cayley-Menger type formula for the generalized earth mover's distance

The earth mover's distance (EMD), also known as the 1-Wasserstein metric, measures the minimum amount of work required to transform one probability distribution into another. The EMD can be naturally generalized to measure the "distance" between any number (say $d$) of distributions. In previous work (2021), we found a recursive formula for the expected value of the generalized EMD, assuming the uniform distribution on the standard $n$-simplex. This recursion, however, was computationally expensive, requiring $\binom{d+n}{d}$ many iterations. The main result of the present paper is a nonrecursive formula for this expected value, expressed as the integral of a certain polynomial of degree at most $dn$. As a secondary result, we resolve an unanswered problem by giving a formula for the generalized EMD in terms of pairwise EMDs; this can be viewed as an analogue of the Cayley-Menger determinant formula that gives the hypervolume of a simplex in terms of its edge lengths.

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arXiv2024

Limits and Periodicity of Metamour $2$-Distance Graphs

Given a finite simple graph $G$, let $\operatorname{M}(G)$ denote its 2-distance graph, in which two vertices are adjacent if and only if they have distance 2 in $G$. In this paper, we consider the periodic behavior of the sequence $G, \operatorname{M}(G), \operatorname{M}^2(G), \operatorname{M}^3(G), \ldots$ obtained by iterating the 2-distance operation. In particular, we classify the connected graphs with period 3, and we partially characterize those with period 2. We then study two families of graphs whose 2-distance sequence is eventually periodic: namely, generalized Petersen graphs and complete $m$-ary trees. For each family, we show that the eventual period is 2, and we determine the pre-period and the two limit graphs of the sequence.

math.CO↗

arXiv2024

The Demazure product extended to biwords

The symmetric group $\mathfrak{S}_n$ (and more generally, any Coxeter group) admits an associative operation known as the Demazure product. In this paper, we first extend the Demazure product to the (infinite) set of all biwords on $\{1, \ldots, n\}$, or equivalently, the set of all $n \times n$ nonnegative integer matrices. We define this product diagrammatically, via braid-like graphs we call kelp beds, since they significantly generalize the seaweeds introduced by Tiskin (2015). Our motivation for this extended Demazure product arises from optimization theory, in particular the semigroup of all $(n+1) \times (n+1)$ simple nonnegative integer Monge matrices equipped with the distance (i.e., min-plus) product. As our main result, we show that this semigroup of Monge matrices is isomorphic to the semigroup of biwords equipped with the extended Demazure product. We exploit this isomorphism to write down generating functions for the growth series of the Monge matrices with respect to certain natural matrix norms.

math.CO↗

arXiv2023

Graphical methods and rings of invariants on the symmetric algebra

Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of $G$-invariant polynomial functions on the space $\mathcal{P}^m(V)$ of degree-$m$ homogeneous polynomial functions on $V$. In this paper, we replace $\mathcal{P}^m(V)$ with the full polynomial algebra $\mathcal{P}(V)$. As a result, the invariant ring is no longer finitely generated. Hence instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when $G$ is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of $G$ is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on $\mathcal{P}(V)$. We conclude with examples using our graphical notation, several of which recover classical results.

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arXiv2023

The structure and normalized volume of Monge polytopes

A matrix $C$ has the Monge property if $c_{ij} + c_{IJ} \leq c_{Ij} + c_{iJ}$ for all $i < I$ and $j < J$. Monge matrices play an important role in combinatorial optimization; for example, when the transportation problem (resp., the traveling salesman problem) has a cost matrix which is Monge, then the problem can be solved in linear (resp., quadratic) time. For given matrix dimensions, we define the Monge polytope to be the set of nonnegative Monge matrices normalized with respect to the sum of the entries. In this paper, we give an explicit description and enumeration of the vertices, edges, and facets of the Monge polytope; these results are sufficient to construct the face lattice. In the special case of two-row Monge matrices, we also prove a polytope volume formula. For symmetric Monge matrices, we show that the Monge polytope is a simplex and we prove a general formula for its volume.

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arXiv2023

Stanley decompositions of rings of invariants and certain highest weight Harish-Chandra modules

The first half of this paper is largely expository, wherein we present a systematic combinatorial approach to the theory of polynomial (semi)invariants and multilinear invariants of several vectors and covectors, for the classical groups. This culminates in a graphical description of graded linear bases. By applying well-known results of lattice path combinatorics to Weyl's fundamental theorems of classical invariant theory, we write down Stanley decompositions and Hilbert-Poincare series in terms of families of non-intersecting lattice paths, enumerated with respect to certain corners. In the second half of the paper, we revisit the (semi)invariants in the first half as a special case of a much broader phenomenon. On one hand, polynomial invariants of a group $H$ can be generalized to modules of covariants, i.e., $H$-equivariant polynomial functions between $H$-modules. On the other hand, from the perspective of Roger Howe's theory of dual pairs, these modules of covariants can be viewed as infinite-dimensional simple $(\mathfrak{g}, K)$-modules. This suggests an expanded program in which our goal is to apply combinatorial techniques involving lattice paths in order to write down Hilbert series for arbitrary unitarizable highest-weight $(\mathfrak{g},K)$-modules. As a preview of future work in this program, we present examples showing how modules of covariants -- even those which are not Cohen-Macaulay, and therefore which we would not expect to be combinatorially nice -- can be decomposed in terms of lattice paths. We also extend these methods beyond the classical groups.

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arXiv2023

The sum of all width-one matrices

A nonnegative integer matrix is said to be width-one if its nonzero entries lie along a path consisting of steps to the south and to the east. These matrices are important in optimal transport theory: the northwest corner algorithm, for example, takes supply and demand vectors and outputs a width-one matrix. The problem in this paper is to write down an explicit formula for the sum of all width-one matrices (with given dimensions $n \times n$ and given sum $d$ of the entries). We prove two strikingly different formulas. The first, a ${}_4 F_3$ hypergeometric series with unit argument, is obtained by applying the Robinson-Schensted-Knuth correspondence to the width-one matrices; the second is obtained via Stanley-Reisner theory. Computationally, our two formulas are complementary to each other: the first formula outperforms the second if $d$ is fixed and $n$ increases, while the second outperforms the first if $n$ is fixed and $d$ increases. We also show how our result yields a new non-recursive formula for the mean value of the discrete earth mover's distance (i.e., the solution to the transportation problem), whenever the cost matrix has the Monge property.

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