arXiv subjects

Bridget Eileen Tenner

Publications and source records attributed to Bridget Eileen Tenner.

At least 19 recordsLinked to original sources

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

Cyclomatic numbers and permutations

We show that several apparently different aspects of a permutation are all tied to a single quantity, the cyclomatic number of its inversion graph. Every reduced word for the permutation orders the edges of the inversion graph one at a time, with the edges from first-occurrence letters forming a spanning forest and the edges from repeated letters accounting for the rest; the number of repeated letters is therefore the cyclomatic number. The excess of permutation cycles over sum components is also at most this quantity, and it follows that the gap between the Coxeter and reflection lengths is at least the cyclomatic number and at most twice it. When the inversion graph is a forest, these results unify classical characterizations of the boolean permutations due to Edelman, to Tenner, and to Petersen and Tenner. We also give a new proof that every connected acyclic inversion graph is a caterpillar.

math.CO↗

arXiv2026

Forest webs and pattern avoidance

In a recent preprint, Mike Cummings showed that the smooth components of suitably parametrized Springer fibers are in bijection with contracted, fully reduced Plücker degree-two $\mathfrak{sl}_r$-webs of standard type and that are forests. He showed these are enumerated by sequence A116731 in the OEIS, which is equinumerous with permutations avoiding the patterns {321,2143,3124}. Cummings posed the problem of strengthening this enumerative result by finding a bijection between these webs and a collection of pattern-avoiding permutations. Here we solve this problem, although notably not with the collection of patterns that Cummings had proposed. Rather, we give a bijection between this class of webs and permutations avoiding the patterns {132,4321,3214}.

math.CO↗

arXiv2026

Cyclic sieving for a class of rectangular domino tableaux

The cyclic sieving phenomenon (CSP) provides valuable data about symmetry classes of cyclic actions, and has applications to representation theory. In this paper, we enumerate domino tableaux of shape 2-by-n, and use this result to prove a new CSP on these objects. We then enumerate the rectangular domino tableaux of any dimensions, and conjecture a more general CSP on rectangular domino tableaux. As a consequence of the enumerative results, we obtain several identities involving Fibonacci and Catalan numbers.

math.CO↗

arXiv2026

Pattern expansions of permutation statistics

We study the expansions of permutation statistics in the basis of functions counting occurrences of a fixed pattern in a permutation. We show the finiteness of these pattern expansions for a class of permutation statistics including the higher moment statistics, generalizing a result of Berman and Tenner. We also give a combinatorial criterion for the positivity of pattern expansions. Using this criterion, we show that the pattern expansion of the number of reduced words of a permutation is positive and give an enumerative interpretation for the coefficients.

math.CO↗

arXiv2025

On partitions associated with elementary symmetric polynomials

The elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These elementary symmetric partition functions have been studied before, and are related to plethysm. In this note, we study properties of the elementary symmetric partition functions, particularly related to injectivity and the number of parts appearing in their image partitions.

math.CO↗

arXiv2025

Majority relations for Condorcet domains of tiling type

Condorcet domains are subsets of permutations arising in voting theory: regarding their permutations as preference orders on a list of candidates, one avoids Condorcet's paradox when aggregating the preferences via a simple majority relation. We use poset theory to show that, for the subclass of Condorcet domains of tiling type, the majority rule has stronger properties. We then develop techniques to predict the majority rule explicitly for the uniform vote tally on Condorcet domains of tiling type, and apply this to several well-known examples.

math.CO↗

arXiv2025

Two-Term Polynomial Identities

We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{σ(1)} \cdots x_{σ(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $σ$ not fixing 1 or $n$ are eventually commutative in the sense that the equality $x_1\cdots x_k = x_{τ(1)} \cdots x_{τ(k)}$ holds for $k$ large enough and all permutations $τ\in S_k$. Calling the minimal such $k$ the degree of eventual commutativity, we prove that $k$ is never more than $2n-3$, and that this bound is sharp. For various natural examples, we prove that $k$ can be taken to be $n+1$ or $n+2$. In the case when $q \ne 1$, we establish that the algebra must be nilpotent. We, moreover, demonstrate that if an algebra is eventually commutative of arbitrary characteristic, then it has a finite basis of its polynomial identities, thus confirming the Specht conjecture in this particular case.

math.RA↗

arXiv2024

Between weak and Bruhat: the middle order on permutations

We define a partial order $\mathcal{P}_n$ on permutations of any given size $n$, which is the image of a natural partial order on inversion sequences. We call this the ``middle order''. We demonstrate that the poset $\mathcal{P}_n$ refines the weak order on permutations and admits the Bruhat order as a refinement, justifying the terminology. These middle orders are distributive lattices and we establish some of their combinatorial properties, including characterization and enumeration of intervals and boolean intervals (in general, or of any given rank), and a combinatorial interpretation of their Euler characteristic. We further study the (not so well-behaved) restriction of this poset to involutions, obtaining a simple formula for the Möbius function of principal order ideals there. Finally, we offer further directions of research, initiating the study of the canonical Heyting algebra associated with $\mathcal{P}_n$, and defining a parking function analogue of $\mathcal{P}_n$.

math.CO↗

arXiv2024

Prism permutations in the Bruhat order

The boolean elements of a Coxeter group have been characterized and shown to possess many interesting properties and applications. Here we introduce "prism permutations," a generalization of those elements, characterizing the prism permutations equivalently in terms of their reduced words and in terms of pattern containment. As part of this work, we introduce the notion of "calibration" to permutation patterns.

math.CO↗

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

On the Lucky and Displacement Statistics of Stirling Permutations

Stirling permutations are parking functions, and we investigate two parking function statistics in the context of these objects: lucky cars and displacement. Among our results, we consider two extreme cases: extremely lucky Stirling permutations (those with maximally many lucky cars) and extremely unlucky Stirling permutations (those with exactly one lucky car). We show that the number of extremely lucky Stirling permutations of order $n$ is the Catalan number $C_n$, and the number of extremely unlucky Stirling permutations is $(n-1)!$. We also give some results for luck that lies between these two extremes. Further, we establish that the displacement of any Stirling permutation of order $n$ is $n^2$, and we prove several results about displacement composition vectors. We conclude with directions for further study.

math.CO↗

arXiv2024

Runs and RSK tableaux of boolean permutations

We define and construct the "canonical reduced word" of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the "run" statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.

math.CO↗

arXiv2024

The clairvoyant maître d'

In this paper we study a variant of the Malicious Maître d' problem. This problem, attributed to computer scientist Rob Pike in Peter Winkler's book "Mathematical Puzzles: A Connoisseur's Collection", involves seating diners around a circular table with napkins placed between each pair of adjacent settings. The goal of the maître d' is to seat the diners in a way that maximizes the number of diners who arrive at the table to find the napkins on both the left and right of their place already taken by their neighbors. Previous work described a seating algorithm in which the maître d' expects to force about 18% of the diners to be napkinless. In this paper, we show that if the maître d' learns each diner's preference for the right or left napkin before they are placed at the table, this expectation jumps to nearly $1/3$ (and converges to $1/3$ as the table size gets large). Moreover, our strategy is optimal for every sequence of diners' preferences.

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

Discrete geometry for electoral geography

"Compactness," or the use of shape as a proxy for fairness, has been a long-running theme in the scrutiny of electoral districts; badly-shaped districts are often flagged as examples of the abuse of power known as gerrymandering. The most popular compactness metrics in the redistricting literature belong to a class of scores that we call contour-based, making heavy use of area and perimeter. This entire class of district scores has some common drawbacks, outlined here. We make the case for discrete shape scores and offer two promising ideas: a cut score and a spanning tree score. We use recent United States redistricting history as a source of examples. No shape metric can work alone as a seal of fairness, but we argue that discrete metrics are better aligned both with the grounding of the redistricting problem in geography and with the computational tools that have recently gained significant traction in the courtroom.

physics.soc-ph↗

arXiv2023

Mesas of Stirling permutations

Given a Stirling permutation w, we introduce the mesa set of w as the natural generalization of the pinnacle set of a permutation. Our main results characterize admissible mesa sets and give closed enumerative formulas in terms of rational Catalan numbers by providing an explicit bijection between mesa sets and rational Dyck paths.

math.CO↗

arXiv2023

Extensions of Hitomezashi Patterns

Hitomezashi, a form of traditional Japanese embroidery, gives rise to intricate arrangements of axis-parallel unit-length stitches in the plane. Pete studied these patterns in the context of percolation theory, and the first two authors recently investigated additional structural properties of them. In this paper, we establish several optimization-style results on hitomezashi patterns and provide a complete classification of "long-stitch" hitomezashi patterns in which stitches have length greater than 1. We also study variants in which stitches can have directions not parallel to the coordinate axes.

math.CO↗