The 1-box pattern on pattern avoiding permutations

TL;DR

This paper investigates the distribution of the 1-box pattern in 132-avoiding permutations, deriving a generating function, exploring Fibonacci connections, and counting permutations with specific pattern occurrences.

Contribution

It introduces a two-variable generating function for the 1-box pattern in 132-avoiding permutations and analyzes its coefficients, linking to Fibonacci numbers and counting specific permutation classes.

Findings

Derived a two-variable generating function for the pattern

Linked pattern coefficients to Fibonacci numbers

Counted permutations with two and three pattern occurrences

Abstract

This paper is continuation of the study of the 1-box pattern in permutations introduced by the authors in \cite{kitrem4}. We derive a two-variable generating function for the distribution of this pattern on 132-avoiding permutations, and then study some of its coefficients providing a link to the Fibonacci numbers. We also find the number of separable permutations with two and three occurrences of the 1-box pattern.