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arXiv · 2402.05600

Networks bijective to permutations

Keiichi Shigechi
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Abstract

We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation through a Dyck tiling on a ribbon. We introduce a new combinatorial object similar to a tree-like tableau, which we call a forest. A forest is shown to give a permutation, and be bijective to a network corresponding to the inverse of the permutation. We show that the poset of networks is a finite graded lattice and admits an $EL$-labeling. By use of this $EL$-labeling, we show the lattice is supersolvable and compute the M\"obius function of an interval of the poset.

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Keiichi Shigechi. 2024-02-08. Networks bijective to permutations. https://arxiv.org/abs/2402.05600

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