Toppleable Permutations, Excedances and Acyclic Orientations [article]

Arvind Ayyer, Daniel Hathcock, Prasad Tetali
2020 arXiv   pre-print
Recall that an excedance of a permutation π is any position i such that π_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,..., ⌊ (n-1)/2 ⌋}. Additionally, we show that the above is also the
more » ... mber of acyclic orientations with unique sink (AUSOs) of the complete bipartite graph K_⌈ n/2 ⌉, ⌊ n/2 ⌋ + 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.
arXiv:2010.11236v2 fatcat:loqxcyumcvgsfbrg2x6etbx4gm