## Toppleable Permutations, Ursell Functions and Excedances

Series
Combinatorics Seminar
Time
Friday, October 16, 2020 - 10:00am for 1 hour (actually 50 minutes)
Location
Speaker
Arvind Ayyer – Indian Institute of Science, Bengaluru, India – arvind@iisc.ac.in
Organizer

Recall that an excedance of a permutation $\pi$ is any position $i$
such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and
Propp (arXiv:1612.06816) on sorting using toppling, we say that
a permutation is toppleable if it gets sorted by a certain sequence of
toppling moves. For the most part of the talk, we will explain the
main ideas in showing that the number of toppleable permutations on n
letters is the same as those for which excedances happen exactly at
$\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Time permitting, we will give
some ideas showing that this is also the number of acyclic
orientations with unique sink (also known as the Ursell function) of the
complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$.

This is joint work with D. Hathcock (CMU) and P. Tetali (Georgia Tech).