Institute for Computational and Experimental Research in Mathematics (ICERM)

Abstract

For Undergraduate Researchers - come to the 11th floor of 121 South Main Street to check in prior to breakfast on the 10th floor.

Abstract

Geometric combinatorics is an area of mathematics concerned with counting properties of geometric objects described by a finite set of building blocks. Polytopes are geometric objects that can be formed by taking the convex hull of finitely many points. When extracting combinatorial information from geometric objects, a natural approach is to subdivide the combinatorial object into smaller, more accessible pieces. In the case of polytopes a natural approach is to triangulate the polytope into simplices or parallelepipeds. The Ehrhart polynomial of a lattice polytope P encodes the number of integer lattice points in dilates of P, and the h*-polynomial of P is the numerator of the accompanying generating function. In this talk I will present background on polytopes, lattice-point enumeration, and share my favorite family of polytopes. I will conclude the talk with some open problems.

Abstract

Parking functions have been well-studied in combinatorics. We extend the classical definition of patterns in permutations to parking functions. In particular, we study parking functions that avoid permutations of length 3. A number of well-known combinatorial sequences arise in our analysis, and this talk will highlight several enumeration results and conjectures. This project is joint work with Lara Pudwell(Valparaiso U.)

Abstract

Manifolds are ubiquitous in modern mathematics, from the familiar low dimensional examples of curves and surfaces in calculus to higher dimensional abstract examples in geometry, physics, data science, and beyond. They come in a bewildering variety - with the basic question of "what kinds of manifolds are possible" providing a powerful guiding light in topology. While the 19th century witnessed a near-complete understanding of the 1 and 2 dimensional cases, much progress during the 20th century in 3 dimensions was guided by a conjecture of Poincare, first formulated in 1904. Poincare's conjecture - essentially that simple 3 dimensional spaces can be probed effectively using 1-dimensional loops - proved much more difficult than originally hoped, remaining unsolved for nearly 100 years. Following a century of work, its eventual resolution by Perelman in 2002 provided a new and powerful tool - called Geometrization - to the study of all 3-dimensional spaces. And while the arguments involved get quite technical, the big-picture story is a beautiful interplay of shape, symmetry and geometry which deserves to be more widely known. My goal in this talk is to give an overview of this exciting story lying at the heart of modern topology, from what was asked to what the mathematical community has learned.

Abstract

The lattice of noncrossing partitions began as a combinatorial curiosity in the early 1970s, but is now connected to several modern areas of mathematics, including parking functions and the combinatorics of reflection groups. In this talk I will describe some work of mine with Jon McCammond in which we provide a new connection between noncrossing partitions and the topology of complex polynomials.

Abstract

We recall that a permutation of length n is said to be a flattened partition if the leading terms of maximal chains of ascents (called runs) are in increasing order. In this talk, we define an analogous set of objects which we call flattened parking functions: a subset of parking functions for which the leading terms of maximal chains of weak ascents (also called runs) are in weakly increasing order. We say that a parking function is flattened if its runs satisfy this condition. We show that the number of flattened parking functions with one run are enumerated by the Catalan numbers, as they are in bijection with nondecreasing parking functions. For n ≤8, where there are at most four runs, we give data for the number of flattened parking functions and it remains an open problem to give formulas for their enumeration in general. We then specialize to a subset of flattened parking functions that we call flattened S-insertion parking functions. These can be obtained by inserting all numbers of a multiset Swhose elements are in [n] := {1, 2, . . . , n}, into a permutation of [n] and checking that the result is flattened. We provide bijections between flattened S-insertion parking functions and flattened S′-insertion parking functions, where Sand S′ have certain relations. We then specialize to the case S:= r, the multiset with r ones, and we establish a bijection between flattened r-insertion parking functions and set partitions of [n +r] with the first r integers in different subsets. As a corollary, we establish that flattened r-insertion parking functions are enumerated by the r-Bell numbers. We also construct exponential generating functions of these enumerations and give the closed differential form. These results generalize the work of Nabawanda, Rakotondrajao, Bamunoba and Beyene, Mantaci on flattened partitions.

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Abstract

Parking functions are a well-studied combinatorial object with numerous connections to other areas of math. They can be seen in the maximal chains of the noncrossing partition lattice N Cn, an important partially ordered set (poset) that provides an ordering relation to all noncrossing set partitions on the integers 1 through n. Just as there are (n + 1)(n−1) parking functions of length n, there are (n + 1)(n−1) maximal chains in N Cn+1. Stanley defined a bijection between the two sets in 1996. In 1964, Rota introduced the bond lattice, a subposet derived from a given graph. When the graph is of a certain form, the bond lattice is a subposet of N Cn+1, encoding a subset of the parking functions of length n. Our work builds on the work of a previous REU group, who found that the bond lattices of certain triangulation graphs have the same number of maximal chains as there are ordered cycle decompositions on n integers. Anders and Archer further showed that each ordered cycle decomposition corresponds to a rooted unimodal forest. Although there exist many bijections between parking functions and rooted forests, our work develops a recursive bijection between parking functions of a triangulation graph and unimodal unordered rooted labelled forests.

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Abstract

Given a parking function α= (a1, . . . , an) define the ascent set of αto be the set of i such that ai < ai+1 and the descent set as the set of i such that ai > ai+1. Our work focuses on enumerating parking functions with certain ascent or descent sets in order to refine previous counts of ascents and descents in parking functions. In fact, these descent and ascent sets exhibit a remarkable symmetry, which we prove. Additionally, we investigate a specialization of ordinary parking functions called unit interval parking functions (UPFs), in which each car parks at most one spot from its preferred parking spot. The enumeration of UPFs with fixed ascent or descent sets is particularly nice. For example, we give a new, independent proof that UPFs are enumerated by the Fubini numbers via a bijection with Fubini rankings. Then, we extend this connection between Fubini rankings and unit interval parking functions to r-Fubini rankings through an investigation of UPFs based on the indices of their ascents, descents, and ties.

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Abstract

Given a simple graph G, one can define a hyperplane arrangement called the G-Shi arrangement. The Pak-Stanley algorithm labels the regions of this arrangement with G•-parking functions in a way that some G•-parking functions may appear more than once. These repetitions of Pak-Stanley labels are a topic of interest in the study of G-Shi arrangements and G•-parking functions, as well as the many combinatorial objects they are connected to. The key insight of our work is the introduction of a combinatorial model called the “Three Rows Game”. Analyzing the histories of this game and the ways in which they can induce the same outcomes allows us to completely characterize the repetitions of the Pak-Stanley labels for path, cycle, and star graphs, and make substantial progress towards understanding the repetitions of the Pak-Stanley labels for trees and general graphs.

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Abstract

Displacement in the context of classical parking functions measures the total number of spots passed over when all the cars have parked. We obtain enumerative results for the number of classical and prime parking functions where each car has at most one displacement. Bijections are obtained between these parking functions and types of surjective functions, as well as with the k-faces of the permutohedron. Beyond this flavor of parking functions, we enumerate parking functions where up to three cars are displaced and suggest a direction to generalize these formulae for any number of cars. Moreover, an efficient algorithm is obtained that computes the number of parking functions exhibiting specific displacements

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Abstract

Parking sequences are a generalization of parking functions in which cars can have different lengths and there are as many parking spots as the sum of the car lengths. A list of parking spot preferences is said to be a parking sequence for the given car lengths if all cars are able to park. A parking sequence is said to be permutation invariant if all of its rearrangements are parking sequences. While all parking functions (i.e., parking sequences given cars of unit length) are invariant, this is not the case for parking sequences. The overarching goal of this work is to provide necessary and sufficient conditions for a parking sequence to be invariant. While obtaining a full characterization remains elusive, we do so for a number of foundational cases. Our main result is a concise characterization of minimally invariant car lengths, wherein the only invariant parking sequence is the all ones preference list. We moreover provide a full characterization of invariant parking sequences given two and three cars. Lastly, we give alternate proofs of known results for constant and strictly increasing car lengths and derive certain closure properties of invariant parking sequences. We conclude with a conjecture regarding a Boolean formula for minimally invariant parking sequences with four cars, as well as open questions concerning invariance under subgroups of the permutation group and the computational complexity of characterizing invariant parking sequences in their full generality.

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