Permutations with non-decreasing transposition array and pattern avoidance

TL;DR

This paper explores a bijection between permutations and subexcedant functions via transposition arrays, characterizing permutations with non-decreasing transposition arrays and analyzing pattern avoidance of length 3.

Contribution

It introduces a new characterization of permutations with non-decreasing transposition arrays and relates it to cycle structures and pattern avoidance.

Findings

Identifies anti-exceedance positions through transposition arrays

Provides an expression of the bijection in terms of cycle structure

Studies pattern avoidance of length 3 in this permutation class

Abstract

We give some results about a bijection associating each permutation with a subexcedant function. This function is related to a particular decomposition of the permutation as a product of transpositions and therefore it has been called transposition array in the literature. In particular, we identify anti-exceedance positions of the permutation through its transposition array and we give an expression of the bijection in terms of the cycle structure of the permutation. We give a characterization of a family of permutations having a non-decreasing transposition array and study length 3 pattern avoidance therein.