On the sub-permutations of pattern avoiding permutations

TL;DR

This paper explores the relationships between pattern avoidance in permutations and their sub-structures, using tree representations to analyze enumerative and probabilistic properties, with implications for pattern-search algorithms.

Contribution

It introduces a novel approach linking permutation patterns to binary increasing trees, providing new enumeration and probability results for pattern avoidance and sub-permutations.

Findings

Analyzed how pattern avoidance affects sub-permutation patterns.

Derived formulas for counting permutations with specific sub-pattern constraints.

Studied the probability of pattern containment in random permutations.

Abstract

There is a deep connection between permutations and trees. Certain sub-structures of permutations, called sub-permutations, bijectively map to sub-trees of binary increasing trees. This opens a powerful tool set to study enumerative and probabilistic properties of sub-permutations and to investigate the relationships between 'local' and 'global' features using the concept of pattern avoidance. First, given a pattern {\mu}, we study how the avoidance of {\mu} in a permutation {\pi} affects the presence of other patterns in the sub-permutations of {\pi}. More precisely, considering patterns of length 3, we solve instances of the following problem: given a class of permutations K and a pattern {\mu}, we ask for the number of permutations $\pi \in Av_n(\mu)$ whose sub-permutations in K satisfy certain additional constraints on their size. Second, we study the probability for a generic pattern to be contained in a random permutation {\pi} of size n without being present in the sub-permutations of {\pi} generated by the entry $1 \leq k \leq n$. These theoretical results can be useful to define efficient randomized pattern-search procedures based on classical algorithms of pattern-recognition, while the general problem of pattern-search is NP-complete.