Stirling Pairs of Permutations

Stirling permutations are permutations pi of the multiset {1,1,2,2, horizontal ellipsis ,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, horizontal ellipsis ,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.