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David Garber

Publications and source records attributed to David Garber.

At least 19 recordsLinked to original sources
arXiv2026

On the Schur-positivity of various sets of set partitions

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur positivity of symmetric functions naturally associated with set partitions, with respect to two different notions of descent. In the first case, the Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings. In the second case, the Schur functions correspond to two-rows Young diagrams, and the coefficients are partial sums of associated Bell numbers. A key ingredient of our approach in the second case is the notion of a removable singleton, defined algebraically and shown to admit an equivalent combinatorial interpretation via jeu-de-taquin rectification of skew tableaux. As an application, we establish Schur positivity for various classes of symmetric functions indexed by non-crossing partitions and partitions with a given number of parts. We provide an explicit combinatorial description of the tableaux that contribute to the Schur expansion, and we connects the obtained coefficients to some known integer sequences.

math.CO↗
arXiv2026

Touchard-Riordan Polynomials and Schur-positivity of Set Partitions

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur-positivity of symmetric functions naturally associated with set partitions, with respect to a descent set function that considers i as descent, if i and i+1 share a block in the partition. The Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings.

cs.DM↗
arXiv2025

On saturated triangulation-free convex geometric graphs

A convex geometric graph is a graph whose vertices are the corners of a convex polygon P in the plane and whose edges are boundary edges and diagonals of the polygon. It is called triangulation-free if its non-boundary edges do not contain the set of diagonals of some triangulation of P. Aichholzer et al. (2010) showed that the maximum number of edges in a triangulation-free convex geometric graph on n vertices is ${{n}\choose{2}}-(n-2)$, and subsequently, Keller and Stein (2020) and (independently) Ali et al. (2022) characterized the triangulation-free graphs with this maximum number of edges. We initiate the study of the saturation version of the problem, namely, characterizing the triangulation-free convex geometric graphs which are not of the maximum possible size, but yet the addition of any edge to them results in containing a triangulation. We show that, surprisingly, there exist saturated graphs with only g(n) = O(n log n) edges. Furthermore, we prove that for any $n > n_0$ and any $g(n)\leq t \leq {{n}\choose{2}}-(n-2)$, there exists a saturated graph with n vertices and t edges. In addition, we obtain a complete characterization of all saturated graphs whose number of edges is ${{n}\choose{2}}-(n-1)$, which is 1 less than the maximum.

math.CO↗
arXiv2024

Type-B analogue of Bell numbers using Rota's Umbral calculus approach

Rota used the functional L to recover old properties and obtain some new formulas for the Bell numbers. Tanny used Rota's functional L and the celebrated Worpitzky identity to obtain some expression for the ordered Bell numbers, which can be seen as an evident to the fact that the ordered Bell numbers are gamma-positive. In this paper, we extend some of Rota's and Tanny's results to the framework of the set partitions of Coxeter type B.

cs.DM↗
arXiv2024

Combinatorics of q,r-analogues of Stirling numbers of type B

Stirling number of the first and the second kinds have seen many generalizations and applications in various areas of mathematics. We introduce some combinatorial parameters which realize $q$-analogues and Broder's $r$-variants of Stirling numbers of type $B$ of both kinds, which count signed set partitions and signed permutations respectively. Applications to orthogonality relations and power sums are given.

math.CO↗
arXiv2022

Analytic aspects of $q,r$-analogue of poly-Stirling numbers of both kinds

The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these identities one can find recursions, explicit formulas based on the inclusion-exclusion principle, and also exponential generating functions. These Stirling numbers can be considered as members of a wider family of triangles of numbers that are characterized using results of Comtet and Lancaster. We generalize these theorems, which present equivalent conditions for a triangle of numbers to be a triangle of generalized Stirling numbers, to the case of the $q,r$-poly Stirling numbers, which are $q$-analogues of the restricted Stirling numbers defined by Broder and having a polynomial value appearing in their defining recursion. There are two ways to do this and these ways are related by a nice identity.

math.CO↗
arXiv2020

The Worpitzky identity for the groups of signed and even-signed permutations

The well-known Worpitzky identity provides a connection between two bases of $\mathbb{Q}[x]$: The standard basis $(x+1)^n$ and the binomial basis ${{x+n-i} \choose {n}}$, where the Eulerian numbers for the Coxeter group of type $A$ (the symmetric group) serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types $B$ and $D$ (signed and even-signed permutations groups, respectively) using generating function techniques. Motivated by Foata-Sch\"utzenberger and Rawlings' proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs of this identity and for their $q-$analogues in the Coxeter groups of types $B$ and $D$.

math.CO↗
arXiv2019

Signed partitions - A balls into urns approach

Using Reiner's definition of Stirling numbers of type B of the second kind, we provide a 'balls into urns' approach for proving a generalization of a well-known identity concerning the classical Stirling numbers of the second kind: $x^n=\sum\limits_{k=0}^n{S(n,k)[x]_k}.$

math.CO↗
arXiv2019

Some identities involving second kind Stirling numbers of types $B$ and $D$

Using Reiner's definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $R[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.

math.CO↗
arXiv2018

Property testing and expansion in cubical complexes

We consider expansion and property testing in the language of incidence geometry, covering both simplicial and cubical complexes in any dimension. We develop a general method for passing from an explicit description of the cohomology group, which need not be trivial, to a testability proof with linear ratio between errors. The method is demonstrated by testing functions on 2-cells in cubical complexes to be induced from the edges.

math.CO↗
arXiv2014

On the group of alternating colored permutations

The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ and use this point of view to give another expression for the length function. We also use this covering to lift several known parameters of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ to the group of alternating colored permutations.

math.GR↗
arXiv2013

Double Centralizers of Parabolic Subgroups of Braid Groups

We characterize the double centralizer of all parabolic subgroups of the braid groups. We apply this result to provide a new and potentially more efficient solution to the subgroup conjugacy problem for parabolic subgroups. In the course of the proof we also characterize the centralizer for all parabolic subgroups.

math.GR↗
arXiv2013

Length-based attacks in polycyclic groups

After the Anshel-Anshel-Goldfeld (AAG) key-exchange protocol was introduced in 1999, it was implemented and studied with braid groups and with the Thompson group as its underlying platforms. The length-based attack, introduced by Hughes and Tannenbaum, has been used to extensively study AAG with the braid group as the underlying platform. Meanwhile, a new platform, using polycyclic groups, was proposed by Eick and Kahrobaei. In this paper, we show that with a high enough Hirsch length, the polycyclic group as an underlying platform for AAG is resistant to the length-based attack. In particular, polycyclic groups could provide a secure platform for any cryptosystem based on conjugacy search problem such as non-commutative Diffie-Hellman, ElGamal and Cramer-Shoup key exchange protocols.

math.GR↗
arXiv2013

On the structure of fundamental groups of conic-line arrangements having a cycle in their graph

The fundamental group of the complement of a plane curve is a very important topological invariant. In particular, it is interesting to find out whether this group is determined by the combinatorics of the curve or not, and whether it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we investigate the structure of this fundamental group when the graph of the conic-line arrangement is a unique cycle of length $n$ and the conic passes through all the multiple points of the cycle. We show that if n is odd, then the affine fundamental group is abelian but not conjugation-free. For the even case, if n>4, then using quotients of the lower central series, we show that the fundamental group is not even a direct sum of a free abelian group and free groups.

math.GT↗
arXiv2013

On Left regular bands and real Conic-Line arrangements

An arrangement of curves in the real plane divides it into a collection of faces. In the case of line arrangements, there exists an associative product which gives this collection a structure of a left regular band. A natural question is whether the same is possible for other arrangements. In this paper, we try to answer this question for the simplest generalization of line arrangements, that is, conic--line arrangements. Investigating the different algebraic structures induced by the face poset of a conic--line arrangement, we present two different generalizations for the product and its associated structures: an alternative left regular band and an associative aperiodic semigroup. We also study the structure of sub left regular bands induced by these arrangements. We finish with some chamber counting results for conic--line arrangements.

math.CO↗
arXiv2012

A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties

A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators $x_1, ..., x_n$ and the cyclic relations: $x_{i_k}x_{i_{k-1}} ... x_{i_1} = x_{i_{k-1}} ... x_{i_1} x_{i_k} = ... = x_{i_1} x_{i_k} ... x_{i_2}$ with no conjugations on the generators. We have already proved that if the graph of the arrangement is a disjoint union of cycles, then its fundamental group has a conjugation-free geometric presentation. In this paper, we extend this property to arrangements whose graphs are a disjoint union of cycle-tree graphs. Moreover, we study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We show that these presentations satisfy a completeness property in the sense of Dehornoy, if the corresponding graph of the arrangement has no edges. The completeness property is a powerful property which leads to many nice properties concerning the presentation (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).

math.GT↗
arXiv2011

On the structure of conjugation-free fundamental groups of conic-line arrangements

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkable properties: either it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we first give a complete proof to the following key lemma: if we draw a new line through only one intersection point of a given real line arrangement whose fundamental group is conjugation-free, then the fundamental group of the new arrangement is also conjugation-free. Second, we generalize this lemma to the case of conic-line arrangements. Moreover, we prove that once the graph associated to conic-line arrangements (defined slightly different than the corresponding graph for line arrangements) has no cycles, then the fundamental group of its complement has a conjugation-free geometric presentation and in addition can be written as a direct sum of free groups and a free abelian group. Also, we show that if the graph consists of one cycle, and the conic does not pass through all the multiple points corresponding to the vertices of the cycle, then the fundamental group has a conjugation-free geometric presentation as well. For conclusion, we extend the family of real line arrangements having a conjugation-free geometric presentation (for their fundamental group) by defining the notion of a conjugation-free graph. We also extend this notion to certain families of conic-line arrangements.

math.GT↗