The number of \bar{2}413\bar{5}-avoiding permutations

TL;DR

This paper proves that the counting sequence for permutations avoiding the pattern 24135 is the Invert transform of Bell numbers, using permutation decomposition and known pattern-avoidance results.

Contribution

It establishes a new connection between 24135}-avoiding permutations and Bell numbers through the Invert transform, providing a combinatorial proof.

Findings

Counting sequence is the Invert transform of Bell numbers

Decomposition approach simplifies understanding of pattern-avoiding permutations

Confirms a question posed by R. J. Mathar

Abstract

We answer a question of R. J. Mathar and confirm that the counting sequence for $\bar{2}413\bar{5}$-avoiding permutations is the Invert transform of the Bell numbers. The proof relies on a simple decomposition of these permutations and the known fact that $\bar{2}413$-avoiding permutations are counted by the Bell numbers.