Inversion Generating Functions for Signed Pattern Avoiding Permutations

We consider the classical Mahonian statistics on the set Bn(Σ) of signed permutations in the hyperoctahedral group Bn which avoid all patterns in Σ, where Σ is a set of patterns of length two. In 2000, Simion gave the cardinality of Bn(Σ) in the cases where Σ contains either one or two patterns of length two and showed that |Bn(Σ)| is constant whenever |Σ| = 1, whereas in most but not all instances where |Σ| = 2, |Bn(Σ)| = (n + 1)!. We answer an open question of Simion by providing bijections from Bn(Σ) to Sn+1 in these cases where |Bn(Σ)| = (n + 1)!. In addition, we extend Simion’s work by providing a combinatorial proof in the language of signed permutations for the major index on Bn(21, 2̄1̄) and by giving the major index on Dn(Σ) for Σ = {21, 2̄1̄} and Σ = {12, 21}. The main result of this paper is to give the inversion generating functions for Bn(Σ) for almost all sets Σ with |Σ| 6 2.