Pattern restricted quasi-Stirling permutations

Abstract: We define a variation of Stirling permutations, called quasi-Stirling permutations, to be permutations on the multiset ${1,1,2,2,\ldots, n,n}$ that avoid the patterns 1212 and 2121. Their study is motivated by a known relationship between Stirling permutations and increasing ordered rooted labeled trees. We construct a bijection between quasi-Stirling permutations and the set of ordered rooted labeled trees and investigate pattern avoidance for these permutations.