The $X$-class and almost-increasing permutations

TL;DR

This paper introduces a bijection between X-shape permutations and a set related to sorting, generalizes to almost-increasing permutations, and derives their generating functions and refined enumerations.

Contribution

It establishes a new bijection, defines almost-increasing permutations via forbidden patterns, and provides their generating functions and detailed enumerations.

Findings

Bijection between X-shape permutations and sorting-related permutations

Generating functions for almost-increasing permutations

Refined counts by cycles, fixed points, excedances, inversions

Abstract

In this paper we give a bijection between the class of permutations that can be drawn on an X-shape and a certain set of permutations that appears in [Knuth] in connection to sorting algorithms. A natural generalization of this set leads us to the definition of almost-increasing permutations, which is a one-parameter family of permutations that can be characterized in terms of forbidden patterns. We find generating functions for almost-increasing permutations by using their cycle structure to map them to colored Motzkin paths. We also give refined enumerations with respect to the number of cycles, fixed points, excedances, and inversions.