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 of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2
\rfloor + 1}$.

This is joint work with P. Tetali (GATech) and D. Hathcock (CMU), and is available at arXiv:2010.11236.

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Last updated: 15 Jul 2024