Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3

TL;DR

This paper investigates permutations avoiding specific vincular and covincular patterns of length 3, revealing connections to known sequences like Catalan and Motzkin numbers, and introduces new enumerations linked to combinatorial objects.

Contribution

It provides a comprehensive enumeration of permutations avoiding a vincular and a covincular pattern of length 3, including new sequences and proofs of known results.

Findings

Identified sequences related to Catalan and Motzkin numbers.

Discovered new sequences connected to lattice paths and partitions.

Settled a conjecture on Wilf-equivalence of barred patterns.

Abstract

Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of permutations simultaneously avoiding a vincular and a covincular pattern, both of length 3, with at most one restriction. We see familiar sequences, such as the Catalan and Motzkin numbers, but also some previously unknown sequences which have close links to other combinatorial objects such as lattice paths and integer partitions. Where possible we include a generating function for the enumeration. One of the cases considered settles a conjecture by Pudwell (2010) on the Wilf-equivalence of barred patterns. We also give an alternative proof of the classic result that permutations avoiding 123 are counted by the Catalan numbers.