A recursive bijective approach to counting permutations containing 3-letter patterns

TL;DR

This paper introduces a recursive bijective method to count permutations with specific three-letter patterns, providing combinatorial proofs and explicit formulas for various pattern occurrences.

Contribution

The paper develops a new recursive bijective approach for counting permutations with multiple three-letter patterns, extending known formulas and conjectures.

Findings

Proves Bona's formula for permutations with one 132 pattern

Derives Noonan's formula for permutations with one 123 pattern

Expresses counts of permutations with multiple patterns using ballot numbers and powers of 2

Abstract

We present a method, illustrated by several examples, to find explicit counts of permutations containing a given multiset of three letter patterns. The method is recursive, depending on bijections to reduce to the case of a smaller multiset, and involves a consideration of separate cases according to how the patterns overlap. Specifically, we use the method (i) to provide combinatorial proofs of Bona's formula {2n-3}choose{n-3} for the number of n-permutations containing one 132 pattern and Noonan's formula 3/n {2n}choose{n+3} for one 123 pattern, (ii) to express the number of n-permutations containing exactly k 123 patterns in terms of ballot numbers for k<=4, and (iii) to express the number of 123-avoiding n-permutations containing exactly k 132 patterns as a linear combination of powers of 2, also for k<=4. The results strengthen the conjecture that the counts are algebraic for all k.