arXiv subjects

Joel Brewster Lewis

Publications and source records attributed to Joel Brewster Lewis.

At least 19 recordsLinked to original sources

arXiv2026

Counting factorizations of Singer cycles in linear and unitary groups

We count factorizations of Singer cycles as products of reflections in the families of special and general unitary and linear groups over a finite field. In the case of minimum-length factorizations, the resulting answer is a striking product formula resembling the count for minimum-length factorizations of Coxeter elements into reflections in complex reflections groups. Moreover, for minimum length, the answers for the unitary and linear groups exhibit the phenomenon of Ennola duality, where the number of factorizations in a unitary group over the field $\mathbb{F}_q$ is given by replacing `$q$' with `$-q$' in the corresponding answer for a linear group. We use the character theory of these groups to make this count, and in particular we employ the Deligne--Lusztig theory of characters for finite reductive groups.

math.CO↗

arXiv2026

Global patterns in signed permutations

Global permutation patterns have recently been shown to characterize important properties of a Coxeter group. Here we study global patterns in the context of signed permutations, with both characterizing and enumerative results. Surprisingly, many properties of signed permutations may be characterized by avoidance of the same set of patterns as the corresponding properties in the symmetric group. We also extend previous enumerative work of Egge, and our work has connections to the Garfinkle--Barbasch--Vogan correspondence, the Erdős--Szekeres theorem, and well-known integer sequences.

math.CO↗

arXiv2026

Non-attacking rook placements on crossword grids

We introduce the notion of a non-attacking rook placement on a crossword grid. A crossword grid is a collection of white squares (which comprise across and down words) and black squares (which separate the words), and a complete non-attacking rook placement on such a grid is a subset of white squares which intersects every across and every down word exactly once. We prove an upper bound on the number of rook placements that a general grid can admit. We then study sparse grids in which no two black squares share an edge and show that rook placements on certain sparse grids correspond bijectively to alternating sign matrices with prescribed $-1$ entries. Specializing further to permutation grids, we prove that every permutation grid admits at least one rook placement, and characterize the permutations whose grids admit exactly one placement in terms of the Robinson--Schensted correspondence. Throughout, we pose a variety of conjectures and open questions.

math.CO↗

arXiv2026

The path-missing and path-free complexes of a directed graph

We study two simplicial complexes arising from a directed graph $G = (V, E)$ with two chosen vertices $s$ and $t$: the *path-free complex*, consisting of all subsets $F \subseteq E$ that contain no path from $s$ to $t$, and the *path-missing complex*, its Alexander dual. Using discrete Morse theory, we prove that both complexes have well-behaved homotopy types -- either contractible or homotopy-equivalent to spheres.

math.CO↗

arXiv2026

Enumeration of Nondegenerate $2 \times (k+1) \times k$ Hypermatrices

We consider the problem of enumerating hypermatrices of format $2 \times (k + 1) \times k$ over a finite field that have nonzero hyperdeterminant and whose nonzero entries are restricted to a plane partition. We conjecture an attractive product formula for the enumeration, and prove it in many cases. In general, we show that the enumeration is given (up to a power of $q - 1$) by a polynomial in $q$ with nonnegative integer coefficients, whose value at $q = 1$ enumerates a natural family of three-dimensional rook placements.

math.CO↗

arXiv2026

Garsia--Remmel $q$-rook numbers are not always unimodal

We show by an explicit example that the Garsia--Remmel $q$-rook numbers of Ferrers boards do not all have unimodal sequences of coefficients. This resolves in the negative a question from 1986 by the aforementioned authors.

math.CO↗

arXiv2025

Interval-closed set rowmotion and homomesy on products of two chains

We study rowmotion dynamics on interval-closed sets. Our first main result proves a simplification of the global definition of interval-closed set rowmotion from (Elder, Lafrenière, McNicholas, Striker, and Welch 2024). We then completely describe the orbits of interval-closed set rowmotion on products of two chains $[2]\times[n]$ and use this understanding to prove a homomesy conjecture from (ELMSW 2024) involving the signed cardinality statistic.

math.CO↗

arXiv2024

Princ-wiki-a Mathematica: Wikipedia editing and mathematics

Over the past 20 years, Wikipedia has gone from a rather outlandish idea to a major reference work, with more than 60 million articles across all languages, including nearly 7 million in English [Wiki01]. Around 27,000 of these articles concern mathematics [b], and Wikipedia is the first place that many of us go to learn about a new mathematical idea. In this overview, we will discuss how to go about creating or editing an article on a mathematical subject. (Most of this applies equally to topics from other technical fields.) We will also discuss biographies of mathematicians, articles on mathematical books, and the social dynamics of the Wikipedia editor community.

math.HO↗

arXiv2024

Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks

We find a generating function for interval-closed sets of the product of two chains poset by constructing a bijection to certain bicolored Motzkin paths. We also find a functional equation for the generating function of interval-closed sets of truncated rectangle posets, including the type $A$ root poset, by constructing a bijection to certain quarter-plane walks.

math.CO↗

arXiv2024

$GL_n(\mathbb{F}_q)$-analogues of some properties of $n$-cycles in $\mathfrak{S}_n$

We give analogues in the finite general linear group of two elementary results concerning long cycles and transpositions in the symmetric group: first, that the long cycles are precisely the elements whose minimum-length factorizations into transpositions yield a generating set, and second, that a long cycle together with an appropriate transposition generates the whole symmetric group.

math.GR↗

arXiv2024

Disarray, reduced words, and 321-avoidance in George groups

Previous work has shown that the disarray (or displacement) of an (affine) (signed) permutation is bounded in terms of its Coxeter length. Here, we characterize the permutations for which the bound is sharp in two ways: in terms of a natural property of their reduced words, and by ``globally'' avoiding the pattern 321.

math.CO↗

arXiv2024

Coincidences between intervals in two partial orders on complex reflection groups

In a finite real reflection group, the reflection length of each element is equal to the codimension of its fixed space, and the two coincident functions determine a partial order structure called the absolute order. In complex reflection groups, the reflection length is no longer always equal to the codimension of fixed space, and the two functions give rise to two different partial orders on the group. We characterize the elements $w$ in the combinatorial family $G(m, p, n)$ of complex reflection groups for which the intervals below $w$ in these two posets coincide. We also explore the relationship between this property and other natural properties of elements in complex reflection groups; some general theory of posets arising from subadditive functions on groups; and the particular case of subadditive functions on the symmetric group.

math.CO↗

arXiv2023

Bargain hunting in a Coxeter group

Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost function on transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection $t$ be the distance between the integers transposed by $t$ in the combinatorial representation of the group (à la Eriksson and Eriksson). Arbitrary group elements then have a well-defined cost, obtained by minimizing the sum of the transposition costs among all factorizations of the element. We show that the cost of arbitrary elements can be computed directly from the elements themselves using a simple, intrinsic formula.

math.CO↗

arXiv2023

Hurwitz numbers for reflection groups III: Uniform formulas

We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the culmination of a series of three.

math.CO↗

arXiv2023

Combinatorial formulas for shifted dual stable Grothendieck polynomials

The $K$-theoretic Schur $P$- and $Q$-functions $GP_λ$ and $GQ_λ$ may be concretely defined as weight generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of stable Grothendieck polynomials, and were introduced by Ikeda and Naruse for applications in geometry. Nakagawa and Naruse specified families of dual $K$-theoretic Schur $P$- and $Q$-functions $gp_λ$ and $gq_λ$ via a Cauchy identity involving $GP_λ$ and $GQ_λ$. They conjectured that the dual power series are weight generating functions for certain shifted plane partitions. We prove this conjecture. We also derive a related generating function formula for the images of $gp_λ$ and $gq_λ$ under the $ω$ involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author. Using these results, we verify a conjecture of Ikeda and Naruse that the $GQ$-functions are a basis for a ring.

math.CO↗

arXiv2023

In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants

We prove that two reflection factorizations of a parabolic quasi-Coxeter element in a finite Coxeter group belong to the same Hurwitz orbit if and only if they generate the same subgroup and have the same multiset of conjugacy classes. As a lemma, we classify the finite Coxeter groups for which every reflection generating set that is minimal under inclusion is also of minimum size.

math.CO↗

arXiv2022

Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements

We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections $\operatorname{Red}_W(g)$ of reduced reflection factorizations of $g$ and $\operatorname{RGS}(W,g)$ of the relative generating sets of $g$. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size $\#\operatorname{Red}_W(g)$ with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-$0$ Hurwitz numbers.

math.CO↗

arXiv2022

A Random Card Shuffling Process

Consider a randomly shuffled deck of $2n$ cards with $n$ red cards and $n$ black cards. We study the average number of moves it takes to go from a randomly shuffled deck to a deck that alternates in color by performing the following move: If the top card and the bottom card of the deck differ in color place the top card at the bottom of the deck, otherwise, insert the top card randomly in the deck. We use tools from combinatorics, probability, and linear algebra to model this process as a finite Markov chain.

math.PR↗