On pattern avoiding indecomposable permutations

TL;DR

This paper classifies indecomposable permutations avoiding certain patterns and provides enumeration formulas, advancing understanding of their structure and distribution, with applications to permutation pattern avoidance and combinatorial statistics.

Contribution

It offers a complete classification of pattern-avoiding indecomposable permutations and recursive enumeration formulas, including bijective proofs and descent statistic analysis.

Findings

Classified indecomposable permutations avoiding patterns of length 3 or 4.

Derived recursive formulas for $12 ightarrow k$-avoiding indecomposables.

Connected descent statistics with pattern avoidance results.

Abstract

Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3. Further, we provide a recursive formula for enumerating $12\cdots k$-avoiding indecomposable permutations for $k\geq 3$. Several of our results involve the descent statistic. We also provide a bijective proof of a fact relevant to our studies.