Stirling Pairs of Permutations

Stirling permutations are permutations $$\pi $$ π of the multiset $$\{1,1,2,2,\ldots ,n,n\}$$ { 1 , 1 , 2 , 2 , … , n , n } in which those integers between the two occurrences of an integer are greater than it. We identify a permutation $$\pi $$ π of $$\{1,1,2,2,\ldots ,n,n\}$$ { 1 , 1 , 2 , 2 , … , n , n } as a pair of permutations $$(\pi _1,\pi _2)$$ ( π 1 , π 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.