Triangle Lectures in Combinatorics

These are a series of combinatorial workshops, held once per semester, each on a Saturday. The series began with its first meeting in spring 2010. They rotate among the universities in and around the Research Triangle. Participants come from numerous colleges and universities within a few hours drive, and some from even farther away. These workshops are funded by the National Science Foundation, in particular enabling us to bring in four exciting speakers to give one hour talks each time as well as funding travel expenses for participants.

TLC steering committee: Nathan Reading (NCSU), Ricky Liu (NCSU), Ezra Miller (Duke University), and Gabor Pataki (UNC Chapel Hill).

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.

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

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.

Past meetings:

Twentieth meeting: Saturday, November 14, 2020 at UNC Greensboro (online)
Speakers: Sara Billey (University of Washington, Seattle), Hao Huang (Emory University), Igor Pak (University of California, Los Angeles), and Nathan Williams (University of Texas, Dallas)
Organizing Committee: Clifford Smyth (UNC Greensboro), Thomas McConville (Kennesaw State U.), Cynthia Vinzant (NCSU)
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Nineteenth meeting: Saturday, February 29, 2020 at UNC Charlotte
Speakers: Margaret Bayer (U Kansas), Michel Goemans (MIT), Svetlana Poznanović (Clemson), Cynthia Vinzant (NCSU)
Organizing Committee: Gabor Hetyei (UNC Charlotte), Ricky Liu (NCSU), Gabor Pataki (UNC Chapel Hill)
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Eighteenth meeting: Saturday, November 16, 2019 at NCSU
Speakers: Georgia Benkart (University of Wisconsin at Madison), Javier Peña (Carnegie Mellon University), Margaret Readdy (University of Kentucky), Alexander Yong (University of Illinois at Urbana-Champaign)
Organizing Committee: Patricia Hersh (NCSU), Gabor Hetyei (UNC-Charlotte), Kailash Misra (NCSU), Gabor Pataki (UNC Chapel Hill)
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Seventeenth meeting: Saturday, March 30, 2019 at Wake Forest University
Speakers:  James Haglund (University of Pennsylvania), Adam Marcus (Princeton University), Jennifer Morse (University of Virginia), Nathan Reading (North Carolina State University)
Organizing committee: Sarah Mason (Wake Forest), Ed Allen (Wake Forest), Patricia Hersh (NCSU), Cliff Smyth (UNC Greensboro)
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Sixteenth meeting: November 10, 2018 at UNC Chapel Hill
Speakers: László Babai (University of Chicago), Daniel Bienstock (Columbia), Rebecca Goldin (George Mason University), and Lek-Heng Lim (University of Chicago).
Organizing Committee: Prakash Belkale (UNC Chapel Hill), Gabor Pataki (UNC Chapel Hill), Richard Rimanyi (UNC Chapel Hill), and Seth Sullivant (NCSU).
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Fifteenth meeting: March 26, 2018 at NCSU
Speakers: Richard Kenyon (Brown), Karola Mészáros (Cornell), Gabor Hetyei (UNC-Charlotte), and Seth Sullivant (NCSU).
Organizing Committee: Patricia Hersh (NCSU), Ricky Liu (NCSU), and Cynthia Vinzant (NCSU)
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Fourteenth meeting: November 19, 2016 at NCSU
Speakers: Bruno Benedetti (U. Miami), Maria Chudnovsky (Princeton), Jeffery Lagarias (U. Michigan), Josephine Yu (Georgia Tech).
Organizing Committee: Ricky Liu (NCSU), Seth Sullivant (NCSU), and Cynthia Vinzant (NCSU)
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Thirteenth meeting: Saturday February 27, 2016 at UNC Greensboro
Speakers: Penny Haxell (U. Waterloo), Jeff Kahn (Rutgers), Greta Panova (U. Penn), Peter Winkler (Dartmouth).
Organizing Committee: Clifford Smyth, chair (UNC Greensboro), David Galvin (Notre Dame), and Patricia Hersh (NCSU).
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Twelfth meeting: Saturday October 3, 2015 at Duke
Speakers: Yuliy Baryshnikov (UIUC), Susan Holmes (Stanford), Dana Randall (Georgia Tech), and John Shareshian (Washington University in St. Louis).
Organizing Committee: Sayan Mukherjee, chair (Duke), John Harer (Duke), Patricia Hersh (NCSU), and Mauro Maggioni (Duke)
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Eleventh meeting: February 21, 2015 at NCSU
Speakers: Matthew Baker (Georgia Tech), Henry Cohn (Microsoft New England), Lionel Levine (Cornell), and Anne Schilling (UC Davis).
Organizing Committee: Ricky Liu (NCSU), Seth Sullivant (NCSU), and Cynthia Vinzant (NCSU)
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Tenth meeting: October 4, 2014, at High Point University
Speakers: Richard Ehrenborg (Kentucky), Robert Ghrist (Penn), Luis Serrano (U Quebec a Montreal), and Rekha Thomas (U. Washington).
Organizing Committee: Lindsay Piechnik (chair, High Point), Ed Allen (Wake Forest), Sarah Mason (Wake Forest), Seth Sullivant (NCSU), and Laurie Zack (High Point)
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Ninth meeting: February 22, 2014 at UNC-Chapel Hill.
Speakers: Shankar Bhamidi (UNC-Chapel Hill), Anders Buch (Rutgers), Pablo Parrilo (MIT), Eva Tardos (Cornell)
Organizing committee: Gabor Pataki (UNC-Chapel Hill), Lindsay Piechnik (High Point University), Scott Provan (UNC-Chapel Hill), Richard Rimanyi (UNC-Chapel Hill), Jack Snoeyink (UNC-Chapel Hill)
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Eighth meeting: September 21, 2013 at NCSU.
Speakers: George Andrews (Penn State), Matt Beck (San Francisco State University), Robin Pemantle (University of Pennsylvania), and Victoria Powers (Emory University)
Organizing committee: Michael Singer (NCSU), Carla Savage (NCSU), and Seth Sullivant (NCSU)
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Seventh meeting: February 9, 2013 at Wake Forest University.
Speakers: Louis Billera (Cornell), Rod Canfield (University of Georgia), Matthew Kahle (Ohio State University), Michelle Wachs (University of Miami)
Organizing committee: Sarah Mason (chair, Wake Forest University), Ed Allen (Wake Forest University), Alex Fink, (NCSU), Patricia Hersh (NCSU)
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Sixth meeting: September 22, 2012 at NCSU.
Speakers: Allen Knutson (Cornell), Vin de Silva (Pomona College), Richard Stanley (MIT), Lauren Williams (UC Berkeley).
Organizing committee: Alex Fink (NCSU), Patricia Hersh (NCSU), Carla Savage (NCSU).
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Fifth meeting: February 11, 2012 at Duke University.
Speakers: Alex Fink (NCSU), Sergey Fomin (Michigan), Nets Katz (Indiana University), Isabella Novik (University of Washington).
Organizing Committee: Christine Berkesch (Duke), Sonja Mapes (Duke), Ezra Miller (Duke).
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Fourth meeting: November 5, 2011 at UNC Chapel Hill.
Speakers: Thomas Lam (Michigan), Jesus De Loera (UC Davis), Ezra Miller (Duke), Doron Zeilberger (Rutgers)
Organizing committee: Prakash Belkale (UNC Chapel Hill), Gabor Pataki (UNC Chapel Hill), Robert Proctor (UNC Chapel Hill), Scott Provan (UNC Chapel Hill), Richard Rimanyi(UNC Chapel Hill).
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Third meeting: April 9, 2011 at NCSU.
Speakers: Prakash Belkale (UNC Chapel Hill), Vic Reiner (University of Minnesota), John Stembridge (University of Michigan), Stephanie van Willigenburg (UBC).
Organizing committee: Hoda Bidkhori (NCSU), Alex Fink (NCSU), Patricia Hersh (NCSU), Carla Savage (NCSU).
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Second meeting: September 25, 2010 at Duke.
Speakers: Alexander Barvinok (University of Michigan), Anne Shiu (Duke), Sami Assaf (MIT), Persi Diaconis (Stanford).
Organizing committee: Patricia Hersh (NCSU), Sonja Mapes (Duke), Ezra Miller (Duke).
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First meeting: February 6, 2010 at NCSU.
Speakers: Carla Savage (NCSU), Bernd Sturmfels (UC Berkeley), Ed Swartz (Cornell), Laszlo Szekely (University of South Carolina).
Organizing committee: Patricia Hersh (NCSU), Ezra Miller (Duke), Scott Provan (UNC), Nathan Reading (NCSU).
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