arXiv subjects

Richard Ehrenborg

Publications and source records attributed to Richard Ehrenborg.

At least 19 recordsLinked to original sources

arXiv2026

Volumes of consecutively defined sets

We study a variant of the graph polytopes of a path and of a cycle where we replace the inequality $x_{i} + x_{i+1} \leq 1$ with the two inequalities $(1-\alpha) \cdot x_{i} + \alpha \cdot x_{i+1} \leq \alpha$ for $0 \leq x_{i} \leq \alpha$ and $\alpha \cdot x_{i} + (1-\alpha) \cdot x_{i+1} \leq \alpha$ for $\alpha \leq x_{i} \leq 1$. Using a self-adjoint operator and its eigenvalues we obtain convergent series for their volumes. As a corollary we obtain that the volumes of the set associated to a path on $n$ vertices and the set associated to a cycle on $n+1$ vertices are related by a constant factor of $\alpha$.

math.CO↗

arXiv2025

The toric g-vector of nestohedra

We show the entries of the toric g-vector of any dual simplicial polytope is a nonnegative integer linear combination of its gamma-vector entries. We give combinatorial interpretations of the toric g-vector of three classical polytopes: the associahedron, the cyclohedron and the permutahedron. Each is expressed in terms of 123-avoidance on a set of structures, namely, parking functions, functions and parking trees. Using our notion of parking trees, we extend our combinatorial model to all chordal nestohedra. As a corollary we obtain combinatorial proofs of nonnegativity of the toric g-vector for all of these polytopes.

math.CO↗

arXiv2024

Conjectures for cutting pizza with Coxeter arrangements

We are interested in conjecturing the sign of the pizza quantity P(H,B(a,R)) for the irreducible Coxeter arrangements H of type A_n, where n=2,3 bmod 4, and type D_n, where n is odd. Our approach is to express the pizza quantity in terms of the pizza quantity of subarrangements known as 2-structures, and we obtain the first non-zero term in the multivariate Taylor expansion.

math.CO↗

arXiv2024

Cyclotomic enumeration of polynomials

Using the cyclotomic identity we compute sums over d-tuples of monic polynomials in F_q[x] weighted by the multiplicity of their irreducible factors. As consequences we determine explicit expressions for the number of d-tuples of polynomials such that their greatest common divisor is rth power free. We also compute the number of monic polynomials where the multiplicity of each irreducible factor belongs to the monoid generated by two relatively prime integers.

math.NT↗

arXiv2024

On the structure of the d-indivisible noncrossing partition posets

We study the poset of d-indivisible noncrossing partitions introduced by M\"uhle, Nadeau and Williams. These are noncrossing partitions such that each block and each dual block has cardinality~$1$ modulo $d$. Generalizing the work of Speicher, we introduce a generating function approach to reach new enumerative results and recover some known formulas on the cardinality, the M\"obius function and the rank numbers. We compute the antipode of the Hopf algebra of d-indivisible noncrossing partition posets. Generalizing work of Stanley, we show that the bijection between the maximal chains and d-parking functions introduced by M\"uhle, Nadeau and Williams gives rise to an EL-labeling. Generalizing a construction of Yan, we also introduce d-parking trees which are in bijective correspondence with the maximal chains.

math.CO↗

arXiv2024

Two classes of level Eulerian posets

We present two classes of level Eulerian posets. Both classes contain intervals of rank k+1 whose cd-index is the sum over all cd-monomials w of degree k and the coefficient of the monomial w is r to the power of the number of d's in w. We also show that the order complexes of every interval in the first class are homeomorphic to spheres.

math.CO↗

arXiv2023

Pizza and 2-structures

Let $\mathcal{H}$ be a Coxeter hyperplane arrangement in $n$-dimensional Euclidean space. Assume that the negative of the identity map belongs to the associated Coxeter group $W$. Furthermore assume that the arrangement is not of type $A_1^n$. Let $K$ be a measurable subset of the Euclidean space with finite volume which is stable by the Coxeter group $W$ and let $a$ be a point such that $K$ contains the convex hull of the orbit of the point $a$ under the group $W$. In a previous article the authors proved the generalized pizza theorem: that the alternating sum over the chambers $T$ of $\mathcal{H}$ of the volumes of the intersections $T\cap(K+a)$ is zero. In this paper we give a dissection proof of this result. In fact, we lift the identity to an abstract dissection group to obtain a similar identity that replaces the volume by any valuation that is invariant under affine isometries. This includes the cases of all intrinsic volumes. Apart from basic geometry, the main ingredient is a theorem of the authors where we relate the alternating sum of the values of certain valuations over the chambers of a Coxeter arrangement to similar alternating sums for simpler subarrangements called $2$-structures introduced by Herb to study discrete series characters of real reduced groups.

math.CO↗

arXiv2023

Catalan-Spitzer permutations

We study two classes of permutations intimately related to the visual proof of Spitzer's lemma and Huq's generalization of the Chung-Feller theorem. Both classes of permutations are counted by the Fuss-Catalan numbers. The study of one class leads to a generalization of results of Flajolet from continued fractions to continuants. The study of the other class leads to the discovery of a restricted variant of the Foata--Strehl group action.

math.CO↗

arXiv2023

Optimization Perspectives on Shellsort

Shellsort is a sorting method that is attractive due to its simplicity, yet it takes effort to analyze its efficiency. The heart of the algorithm is the gap sequence chosen a priori and used during sorting. The selection of this gap sequence affects the efficiency of Shellsort, and thus drives both its theoretical and experimental analysis. We contribute to Shellsort by identifying efficient gap sequences based on new parameterized functions. Specifically, a parameter grid-search identifies optimal parameters for different input sizes for sorting by observing minimal overhead in three categories: number of comparisons, number of exchanges, and running time. We report that our method finds sequences that outperform state-of-the-art gap sequences concerning the number of comparisons for chosen small array sizes. Additionally, our function-based sequences outperform the running time of the Tokuda sequence for chosen large array sizes. However, no substantial improvements were observed when minimizing the number of exchanges.

cs.DS↗

arXiv2022

A generalization of combinatorial identities for stable discrete series constants

This article is concerned with the constants that appear in Harish-Chandra's character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra's work the only information we have about these constants is that they are uniquely determined by an inductive property. Later Goresky-Kottwitz-MacPherson and Herb gave different formulas for these constants. In this article we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel. We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Appendix A of the Goresky-Kottwitz-MacPherson paper. In Appendix B we extend the notion of $2$-structures (due to Herb) to pseudo-root systems.

math.CO↗

arXiv2022

Sharing pizza in n dimensions

We introduce and prove the $n$-dimensional Pizza Theorem: Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{R}^{n}$. If $K$ is a measurable set of finite volume, the {pizza quantity} of $K$ is the alternating sum of the volumes of the regions obtained by intersecting $K$ with the arrangement $\mathcal{H}$. We prove that if $\mathcal{H}$ is a Coxeter arrangement different from $A_{1}^{n}$ such that the group of isometries $W$ generated by the reflections in the hyperplanes of $\mathcal{H}$ contains the map $-\mathrm{id}$, and if $K$ is a translate of a convex body that is stable under $W$ and contains the origin, then the pizza quantity of $K$ is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of $\mathcal{H}$ that we call the {even restricted arrangement}. More generally, we prove that for a class of arrangements that we call {even} (this includes the Coxeter arrangements above) and for a {sufficiently symmetric} set $K$, the pizza quantity of $K+a$ is polynomial in $a$ for $a$ small enough, for example if $K$ is convex and $0\in K+a$. We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at $a$ having radius $R\geq\|a\|$ vanishes for a Coxeter arrangement $\mathcal{H}$ with $|\mathcal{H}|-n$ an even positive integer. We also prove the Pizza Theorem for the surface volume: When $\mathcal{H}$ is a Coxeter arrangement and $|\mathcal{H}| - n$ is a nonnegative even integer, for an $n$-dimensional ball the alternating sum of the $(n-1)$-dimensional surface volumes of the regions is equal to zero.

math.CO↗

arXiv2020

Classification of uniform flag triangulations of the boundary of the full root polytope of type $A$

The full root polytope of type $A$ is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations fall naturally into three classes: three in the lex class, three in the revlex class and nine in the Simion class. We also consider a refined face count where we distinguish between forward and backward arrows. We prove the refined face counts only depend on the class of the triangulations. The refined face generating functions are expressed in terms of the Catalan and Delannoy generating functions and the modified Bessel function of the first kind.

math.CO↗

arXiv2019

Bruhat and balanced graphs

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an $R$-labeling, imply the existence of the (non-homogeneous) ${\bf cd}$-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete ${\bf cd}$-index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture nonnegativity of the ${\bf cd}$-index for acyclic digraphs having a balanced linear edge labeling.

math.CO↗

arXiv2019

Some combinatorial identities appearing in the calculation of the cohomology of Siegel modular varieties

In the computation of the intersection cohomology of Shimura varieties, or of the $L^2$ cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always well-understood (see for example the appendix of [8]). We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group. Apart from some introductory remarks about the origin of the identities, our paper is entirely combinatorial and does not require any knowledge of Shimura varieties or of representation theory.

math.CO↗

arXiv2017

Parking cars after a trailer

Recently, the authors extended the notion of parking functions to parking sequences, which include cars of different sizes, and proved a product formula for the number of such sequences. We here give a refinement of that result involving parking the cars after a trailer. The proof of the refinement uses a multi-parameter extension of the Abel--Rothe polynomial due to Strehl.

math.CO↗

arXiv2017

On the powers of the descent set statistic

We study the sum of the $r$th powers of the descent set statistic and how many small prime factors occur in these numbers. Our results depend upon the base $p$ expansion of $n$ and $r$.

math.CO↗

arXiv2017

Box polynomials and the excedance matrix

We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $λ$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference operator applied to the power $x^{m+n}$. Evaluating box polynomials yields a variety of identities involving set partition enumeration. We extend the latter identities using restricted growth words and a new operator called the fast Fourier operator, and consider connections between set partition enumeration and the chromatic polynomial on graphs. We also give connections between the box polynomials and the excedance matrix, which encodes combinatorial data from a noncommutative quotient algebra motivated by the recurrence for the excedance set statistic on permutations.

math.CO↗

arXiv2017

Parking cars of different sizes

We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences.

math.CO↗