Fall 2023

- The Combinatorics Seminar meets on
**Fridays, 3-4pm**, in**Snow 302**. (We may sometimes use Zoom instead.) - Organizers: Mark Denker, Reuven Hodges, and Marge Bayer.
- Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil
- Also consider attending the Graduate Online Combinatorics Colloquium.

**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)**

TBA

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

Last updated 3-Dec-23