Toppleable permutations, excedances and acyclic orientations

Recall that an excedance of a permutation $\pi$ is any position $i$ such $\pi_i > i$. Inspired by the work Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say toppleable if it gets sorted certain sequence toppling moves. One our main results number permutations $n$ letters same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Additionally, show above also acyclic orientations with unique sink (AUSOs) complete bipartite graph $K_{\lceil n/2 \rceil, \rfloor + 1}$. We give formula AUSOs multipartite graphs. conclude observations extremal question Cameron et al. concerning maximizers (the of) orientations, given prescribed vertices edges graph.Mathematics Subject Classifications: 05A19, 05A05, 05C30