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

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.

Equivalence classes of mesh patterns with a dominating pattern

“…After implementing the Second Dominating Pattern Rule we find that there are 39 - (2) 231 -equivalence classes, of mesh patterns where the underlying classical pattern is 21. By Note 3.1 there are therefore exactly 39 -231 -equivalence classes.…”

“…We then prove three propositions called "Dominating Pattern Rules", if the First Dominating Pattern Rule shows that p 1π p 2 then we write p 1 - (1) π p 2 . If a combination of the First and Second Dominating Pattern Rules show that p 1π p 2 then we write p 1 - (2) π p 2 ; similarly we write p 1 -(3) π p 2 if a combination of all three rules shows coincidence. These three equivalence relations are successive refinements ofπ .…”

“…Given two mesh patterns m 1 " pσ, R 1 q and m 2 " pσ, R 2 q, and a dominating classical pattern π " pπ, Hq such that Proof: By Note 3.6 we only need to show that an occurrence of m 1 implies an occurrence of m 2 . We consider taking the first branch of every choice in condition (2). Consider a permutation ω P Avpπq.…”

“…When considering the coincidence classes of Avpt231, p12, Rquq we first apply the two Dominating Pattern rules previously established. Starting from 220 classes, we result in 85 - (1) 231 -equivalence classes reducing to 59 - (2) 231 -equivalence classes, considering mesh patterns with 12 as the underlying classical patterns. However there are 56 -comp 231 -equivalence classes.…”

“…Avoiding pairs of patterns of the same length with certain properties has been studied, Claesson and Mansour [6] considered avoiding a pair of vincular patterns of length 3. Bean, Claesson, and Ulfarsson [2] study avoiding a vincular and a covincular pattern simultaneously in order to achieve several counting results. However, little work has been done on avoiding a mesh pattern and a classical pattern simultaneously.…”