Stirling Pairs of Permutations

Abstract

Stirling permutations are permutations $$\pi $$ of the multiset $$\{1,1,2,2,\ldots ,n,n\}$$ in which those integers between the two occurrences of an integer are greater than it. We identify a permutation $$\pi $$ of $$\{1,1,2,2,\ldots ,n,n\}$$ as a pair of permutations $$(\pi _1,\pi _2)$$ which we call a Stirling pair. We characterize Stirling pairs using the weak Bruhat order and the notion of a 312-avoiding permutation. We give two algorithms to determine if a pair of permutations is a Stirling pair.