Restricted Permutations Enumerated by Inversions

TL;DR

This paper explores the enumeration of indecomposable permutations based on their inversion counts, focusing on pattern restrictions of length up to three, offering new insights beyond traditional size-based enumeration.

Contribution

It provides a comprehensive analysis of permutations counted by inversions with pattern restrictions, extending classical enumeration methods to a new perspective.

Findings

Enumeration formulas for indecomposable permutations by inversions.

Analysis of pattern restrictions of length up to three.

Extension of classical permutation enumeration techniques.

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.