Enumeration of permutations containing a prescribed number of occurrences of a pattern of length 3

TL;DR

This paper introduces a new bijective method to enumerate permutations with a fixed number of occurrences of a length-3 pattern, proving a previously conjectured formula using lattice path techniques.

Contribution

It provides an alternative proof for a known enumeration formula by employing bijections to lattice paths, advancing combinatorial enumeration methods.

Findings

Established a bijective approach to permutation enumeration

Proved a conjectured formula for permutations with fixed pattern occurrences

Connected permutation patterns to lattice path representations

Abstract

We consider the problem of enumerating the permutations containing exactly $k$ occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an alternative approach to the problem, which yields a proof for a formula which so far only was conjectured (by Noonan and Zeilberger). This approach is based on bijections from permutations to certain lattice paths with ``jumps'', which were first considered by Krattenthaler.