Toppleable Permutations, Excedances and Acyclic Orientations

TL;DR

This paper studies toppleable permutations, their connection to excedances and acyclic orientations, providing formulas and characterizations for these combinatorial objects and exploring extremal properties of acyclic orientations.

Contribution

It establishes a novel equivalence between toppleable permutations, excedance positions, and acyclic orientations with a unique sink, along with formulas for counting these orientations.

Findings

Number of toppleable permutations equals those with excedances at specific positions.

Number of acyclic orientations with a unique sink matches toppleable permutations.

Provides formulas for counting acyclic orientations of complete multipartite graphs.

Abstract

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 (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$. We also give a formula for the number of AUSOs of complete multipartite graphs. We conclude with observations on an extremal question of Cameron et al. concerning maximizers of (the number of) acyclic orientations, given a prescribed number of vertices and edges for the graph.