Wave linked partitions and 312-avoiding permutations with primacy being 1

Abstract

In this paper, we mainly study connected non-crossing linked partitions, which are vividly called wave linked partitions, and their relation to 312-avoiding permutations with primacy being 1. Let W(n) be the set of wave linked partitions of {1,2,…,n} and let w(n, k) be the number of wave linked partitions of {1,2,…,n} with wave of length k. We obtain that the enumeration of W(n+1) is counted by the well-known nth Catalan number Cn and we present the recurrence relation of w(n, k). Let P1(n) be the set of 312-avoiding permutations of {1,2,…,n} with primacy being 1. As a main result, we construct an interesting bijection between W(n) and P1(n) by introducing a labeling rule on wave linked partitions. The labeling rule implies that the major index of such kind of permutations can be easily obtained from the labels of certain vertices in the corresponding wave linked partitions.