KU Combinatorics Seminar
Fall 2023

Friday 8/25
Organizational meeting

Friday 9/1
Marge Bayer
Parking Function Polytopes

Friday 9/8
Mark Denker
Promotion Sorting on Posets

Friday 9/15
Dania Morales
Ehrhart Polynomials of Base Polytopes of Lattice Path Matroids

Abstract: Given an integer polytope \(\mathscr{P}\) in \(\mathbb{R}^n\), the Ehrhart polynomial of \(\mathscr{P}\) is a fundamental polytopal invariant that enumerates integer lattice points contained in all nonnegative integer dilations of \(\mathscr{P}\). In 2007, De Loera-Haws-Köppe conjectured that Ehrhart polynomials of matroid base polytopes have all coefficients nonnegative. In 2022, Ferroni showed that this conjecture is not true in general. However, this assertion has led to the major open problem to characterize Ehrhart polynomials for matroid base polytopes. In this talk, we focus on matroid base polytopes of lattice path matroids and give a formula for their Ehrhart polynomials.

Friday 9/22
My Favorite Posets, by Richard Stanley (recorded lecture)

Friday 9/29
Han Yin
Dominating Sets on Cartesian Products of Complete Graphs

Friday 10/6
No seminar

Friday 10/13
Fall Break - no seminar

Friday 10/20
Reuven Hodges
Straightening Fillings of Young Diagrams

Friday 10/27
Matthew Plante (University of Connecticut)
The whirling action on P-partitions and Rowmotion on chain-factor Posets

Abstract: We investigate the dynamics of certain natural actions on labelings of partially ordered sets (posets) and related objections. Rowmotion is an invertible map on order ideals of a poset which has received much attention recently from researchers in dynamical algebraic combinatorics. Of particular interest are the order of the rowmotion map and the homomesy phenomenon. In this talk we look at rowmotion on order ideals of posets of families not yet thoroughly investigated, including fence posets, obtained by arbitrarily ordering each edge in a path graph up or down. These posets are important in the theory of cluster algebras and \(q\)-analogues. The rowmotion orbits of antichains of fence posets can be succinctly visualized using certain tilings of a cylinder.

Another family is the \(\vee\times[k]\) posets, where \(\vee\) is the \(\vee\)-shaped three element poset with two relations and one minimal element, and \([k]\) is a \(k\)-element chain. Here we make an equivariant bijection between rowmotion on order ideals and the whirling action, introduced by Joseph, Propp, and Roby, on \(P\)-partitions of \(\vee\). Finally we investigate whirling proper colorings of path graphs and cycle graphs. This is motivated as a generalization of toggling independent sets of a path graph, which itself has an equivariant bijection to rowmotion on order ideals of a special type of fence poset.

Friday 11/3
Nathan Lesnevich (Washington University in St. Louis)
Splines on Cayley Graphs of Symmetric Groups

Friday 11/10
Aaron Ortiz
Intersecting lattice paths and random weights

Friday 11/17
Garrett Nelson (Kansas State)
Work towards finding a bijection from Deograms to Rational Dyck Paths

Friday 11/24
Thanksgiving - no seminar

Friday 12/1
No seminar

Friday 12/8 (Stop Day)

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated 3-Dec-23