## Combinatorics Seminars

**Spring 2019**

Title: Two problems regarding primitive sets of integers

Speaker: Nathan McNew, Towson

Date and time: Monday, April 15, 4–5pm

Place: Phillips 736

Abstract: A set of integers is primitive if no integer in the set divides another. The prime numbers form one such a set, but there are many others. Let f(n) count the number of primitive subsets of the integers {1, 2, ... n}. Cameron and Erdos showed that 1.55^n < f(n) < 1.59^n and conjectured that f(n) was asymptotic to c^n for some constant c. In 2017 the conjecture was proven Angelo, but his proof is ineffective in the sense that it gives no information about the value of the constant c. We improve Angelo's result by showing how this constant can be computed and obtaining an error term. Then, we use the probabilistic method to construct infinite primitive sets with relatively small gaps between consecutive terms, substantially smaller than is known to hold for the primes. The techniques we present generalize to a variety of related combinatorial questions.

Speaker: Lindsey-Kay Lauderdale, Towson

Date and time: Thursday, April 4, 4-5pm

Place: Rome 352

Abstract: In a popular paper of Cohn, the concept of a covering number of a group was introduced. The

*covering number*of a nontrivial finite group

*G*is the smallest number of proper subgroups of

*G*whose set-theoretic union is

*G*. Covering numbers are the subject of prior research by numerous authors, and in this talk we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. In addition, we will compare our results to some of the well-known results on covering numbers. This is joint work with Kassie Archer and Humberto Bautista Serrano.

Title: Quadrangulated Immersions of Cubic Graphs in the Sphere

Speaker: Lowell Abrams, GWU

Date and time: Thursday, March 28, 4–5pm

Place: Rome 352

Abstract: A quadrangulated immersion of a graph G in a surface S is a drawing of G in S so that each crossing is transversal, each point of crossing is formed by exactly two edges, and each connected region of the complement of G in S is bounded by [portions of] four edges of G. We discuss basic constraints on quadrangulated immersions of cubic graphs in the sphere, and demonstrate various methods of constructing such immersions, including methods for constructing non-isomorphic immersions of the same graph. This is joint work with Yosef Berman, Michael Murphy, and Vance Faber.

Title: Quadrangulated Immersions of Cubic Graphs in the Sphere(CANCELLED/RE-SCHEDULED)

Speaker: Lowell Abrams, GWU

Date and time: Thursday, March 7, 4–5pm

Place: Rome 352

Abstract: A quadrangulated immersion of a graph G in a surface S is a drawing of G in S so that each crossing is transversal, each point of crossing is formed by exactly two edges, and each connected region of the complement of G in S is bounded by [portions of] four edges of G. We discuss basic constraints on quadrangulated immersions of cubic graphs in the sphere, and demonstrate various methods of constructing such immersions, including methods for constructing non-isomorphic immersions of the same graph. This is joint work with Yosef Berman, Michael Murphy, and Vance Faber.

Speaker: Chaim Even Zohar, UC Davis

Date and time: Monday, February 25, 4-5pm

Place: Rome 206

Title: Patterns in Random Permutations

Abstract: Every *k* entries in a permutation can have one of *k*! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size *n*? The distribution of this *k*!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(*n*), while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into *k* asymptotically uncorrelated components of different orders in *n*, that correspond to representations of *S _{k}*. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

Speaker: Jonathan David Farley, Morgan State University

Date and time: Thursday, February 14, 4-5pm

Place: Rome 352

Title: Does Every Infinite Geometric Lattice of Finite Rank Have a Matching? A "Challenging Question" of Björner from 1976

Abstract: At the 1981 Banff Conference on Ordered Sets, Anders Björner asked if every geometric lattice of finite rank greater than 1 had a matching between the points and hyperplanes, a question he called "challenging" in 1976. The matching sought is an injection from points to hyperplanes that matches a point to a hyperplane over it. We answer Björner's question.

Speaker: Valeriu Soltan, George Mason University

Date and time: Thursday, January 31, 4-5pm

Place: Rome 352

Title: Helly-type results on support lines for families of convex ovals

Abstract: We discuss the following new result of combinatorial geometry: a disjoint family of six or more unit circular disks in the plane has a common support line provided every its subfamily of three disks has a common support line.

Speaker: Emanuele Delucchi, University of Fribourg

Date and time: Thursday, January 24, 4-5pm

Place: Rome 352

Title: Stanley-Reisner rings of symmetric simplicial complexes

Abstract: A classical theme in algebraic combinatorics is the study of face rings of finite simplicial complexes (named after Stanley and Reisner, two of the pioneers of this field). In this talk I will examine the case where the simplicial complexes at hand carry a group action and are allowed to be infinite. I will present the foundations of this generalized theory with a special focus on simplicial complexes associated to (semi)matroids, where the associated rings enjoy especially nice algebraic properties. A main motivation for our work comes from the theory of arrangements in abelian Lie groups (e.g., toric and elliptic arrangements), and in particular from the quest of understanding numerical properties of the coefficients of characteristic polynomials and h-polynomials of arithmetic matroids. I will describe our current results in this direction and, time permitting, I will outline some open questions that arise in this new framework. (Joint work with Alessio D'Alì.)

Speaker: Gi-Sang Cheon, Sungkyunkwan University

Date and time: Monday, January 14, 4-5pm

Place: Rome 206

Title: On Riordan graphs

Abstract: In this talk, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this talk is the study of structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which includes a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle.