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Networks bijective to permutations

Keiichi Shigechi

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 Möbius function of an interval of the poset.

Networks bijective to permutations

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 -labeling. By use of this -labeling, we show the lattice is supersolvable and compute the Möbius function of an interval of the poset.
Paper Structure (12 sections, 24 theorems, 60 equations, 13 figures)

This paper contains 12 sections, 24 theorems, 60 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Networks with sources and sinks
  3. Rothe diagrams and networks
  4. A Dyck tiling on a ribbon
  5. Polyominoes and permutations
  6. Polyominoes and networks
  7. Rothe diagrams and networks
  8. A poset of networks
  9. Basic properties of a poset of networks
  10. Forests and Networks
  11. Forests and permutations
  12. Shellability and Möbius functions

Key Result

Theorem 2.4

The cardinality of $\mathcal{N}(n)$ is $n!$, i.e., $|\mathcal{N}_{n}|=n!$.

Figures (13)

  • Figure 2.3: Networks with four points with two sources and two sinks.
  • Figure 3.1: An example of the maximal Dyck tiling on a ribbon. A red line represents a Dyck path which characterizes a Dyck tile.
  • Figure 3.4: An example of a polyomino in $\mathcal{P}$
  • Figure 4.3: A poset $\mathcal{P}(4;\epsilon)$ with $\epsilon=(1,1,-1,-1)$.
  • Figure 5.5: A subposet which has a crossing.
  • ...and 8 more figures

Theorems & Definitions (80)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.4
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • Lemma 2.8
  • proof
  • proof : Proof of Theorem \ref{['thrm:cardN']}
  • Remark 2.9
  • ...and 70 more