## Combinatorics Seminars

**Spring 2019**

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.