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Restricted Permutations Enumerated by Inversions

Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, Jay Pantone

TL;DR

All combinations of permutation patterns of length at most 3.0 are investigated in the style of the seminal paper by Simion and Schmidt.

Abstract

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion and Schmidt, we investigate all combinations of permutation patterns of length at most 3.

Restricted Permutations Enumerated by Inversions

TL;DR

All combinations of permutation patterns of length at most 3.0 are investigated in the style of the seminal paper by Simion and Schmidt.

Abstract

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion and Schmidt, we investigate all combinations of permutation patterns of length at most 3.
Paper Structure (4 sections, 22 theorems, 3 equations)

This paper contains 4 sections, 22 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Single patterns
  3. Several patterns
  4. More than two patterns

Key Result

Lemma 1

Let $\pi$ be a permutation on $n$ elements and $c$ components. Then $\operatorname{inv}(\pi) \geq n - c$.

Theorems & Definitions (22)

  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 12 more