A permutation pattern that illustrates the strong law of small numbers

TL;DR

This paper derives an explicit formula for counting permutations avoiding a specific barred pattern, revealing unique properties of its sequence that initially align with a known sequence but then diverge.

Contribution

It provides the first explicit enumeration formula for permutations avoiding the barred pattern bar{1}43bar{5}2, highlighting novel sequence behavior.

Findings

Derived explicit formula for pattern-avoiding permutations

Identified divergence from a known integer sequence after initial terms

Revealed unique combinatorial properties of the sequence

Abstract

We obtain an explicit formula for the number of permutations of [n] that avoid the barred pattern bar{1}43bar{5}2. A curious feature of its counting sequence, 1, 1, 2, 5, 14, 43, 145, 538, 2194,..., is that the displayed terms agree with A122993 in the On-Line Encyclopedia of Integer Sequences, but the two sequences diverge thereafter.