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- W3093475664 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.Mathematics Subject Classifications: 05A19, 05A05, 05C30" @default.
- W3093475664 created "2020-10-29" @default.
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- W3093475664 date "2022-03-29" @default.
- W3093475664 modified "2023-10-05" @default.
- W3093475664 title "Toppleable permutations, excedances and acyclic orientations" @default.
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- W3093475664 doi "https://doi.org/10.5070/c62156882" @default.
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