# TLC Fall 2022

**Twenty-first meeting: Saturday, December 3, 2022**

** Location: **UNC Greensboro — Sullivan Science Building, SULV 101, and the first-floor lobby.

**Parking:**Parking is free at the McIver, Walker, and Oakland Parking decks!

McIver is the closest, so try that first. If it is full, try Walker next. There is another event parking at McIver that day.

**Speakers: **

Jeremy L. Martin (University of Kansas), Erik Slivken (University of North Carolina Wilmington), Catherine Yan (Texas A&M University), and Yan Zhuang (Davidson College)

** Tentative Conference Schedule** (all times in EST)

(Abstracts can be found below)

09:00 – 10:00am, welcome

10:00 – 11:00am,

**Erik Slivken**,

*Scaling Limits of Restricted Permutations*

11:00 – 11:30am, coffee break

11:30 – 12:30pm,

**Jeremy L. Martin**,

*Unbounded Matroids*

12:30 – 2:30pm, lunch break

02:30 – 3:30pm,

**Yan Zhuang**

*, Shuffle-Compatibility: An Overview*

03:30 – 4:00pm, coffee break

04:00 – 5:00pm,

**Catherine Yan**,

*Parking Functions, Interpolation Polynomials, and Partition Lattice*

** Registration:**To register, please fill out this form.

** Lunch break: **Tate Street has many dining options and is a ten-minute walk from the conference building. If you eat at Boba House, go early as I have always experienced bad wait times, well over 30 minutes.

** Organizing Committee:** Laura Colmenarejo (NCSU), Sarah K. Mason (Wake Forest University), and Clifford Smyth (UNC Greensboro)

### Titles and Abstracts

**Jeremy L. Martin: Unbounded Matroids**

Every matroid *M* gives rise to a *base polytope* whose vertices are the characteristic vectors of its bases. These polytopes are examples of the *generalized permutahedra* studied by Postnikov, Edmonds, and others: their edges are all parallel to differences of standard basis vectors, or equivalently their normal fans coarsen the braid fan. By a theorem of Gel’fand, Goresky, Macpherson and Serganova, matroid polytopes are exactly the generalized permutahedra whose vertices are all 0,1-vectors.

Our goal is to understand the combinatorics-geometry dictionary for 0,1-generalized permutahedra that are not assumed to be bounded. The corresponding “unbounded matroid” consists of a distributive lattice *D*, which encodes the recession cone of the polyhedron, and a submodular rank function with domain *D*. This data is equivalent to a *pregeometry* in the sense of Faigle, and is an instance of the *submodular systems* studied by Fujishige and others. We prove that the simplicial complex generated by bases of an unbounded matroid is shellable. We show every unbounded matroid has a canonical extension to a matroid, and interpret this result both in terms of the corresponding 0,1-polyhedra and in terms of subspace arrangements.

This is joint work with Jonah Berggren (Kentucky) and Jose Samper (Pontificia Universidad Católica de Chile).

**Erik Slivken: Scaling Limits of Restricted Permutations**

Suppose we take a large permutation that is chosen uniformly at random and conditioned to satisfy some restriction. What does this permutation look like? The answer depends on the choice of restriction and how one decides to scale the permutation. We introduce a few scaling limits that prove useful in answering this type of question for a variety of restrictions, especially in the case of pattern-avoiding permutations. We will introduce pattern avoidance and related restrictions to permutations and explore what various scaling limits say about these objects and some associated statistics (like the number of fixed points of the permutation).

**Catherine Yan: Parking Functions, Interpolation Polynomials, and Partition Lattice**

Parking function is an object lying in the center of combinatorics. Originated in the theory of hashing and searching in computer science, parking functions have various generalizations and appear in many discrete and algebraic structures.

In this talk we discuss a special kind of generalization, the vector parking functions, which correspond naturally to Goncarov polynomials, the basis of the solutions of the Goncarov Interpolation Problem in Numerical Analysis. Using the theory of Finite Operator Calculus, we introduce the sequence of delta-Goncarov polynomials, describe their algebraic and analytic properties, and show that any such a polynomial sequence can be realized as a weighted enumerator in the partition lattice. Our result provides an algebraic tool to enumerate combinatorial structures with a linear constraint on their order statistics.

**Yan Zhuang: Shuffle-Compatibility: An Overview**

A permutation statistic st is said to be *shuffle-compatible* if the distribution of st over the set of shuffles of two disjoint permutations *π* and *σ* depends only on st(*π*), st(*σ*), and the lengths of *π* and *σ*. This notion is implicit in Stanley’s work on P-partitions, and was first explicitly studied by Gessel and Zhuang in 2018, who developed a unifying framework for shuffle-compatibility in which quasisymmetric functions play an important role. Since then, shuffle-compatibility has become an active topic of research. In this talk, I will give an overview of the theory of shuffle-compatibility from my joint work with Ira Gessel. Then, I will survey some recent developments including joint work with Jinting Liang and Bruce Sagan on shuffle-compatibility of cyclic permutation statistics (in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions introduced by Adin, Gessel, Reiner and Roichman), as well as advances by Davidson College undergraduates Keegan Stump ’24 and William Clark ’23.