Inversion generating functions for signed pattern avoiding permutations (Q510361)

Summary: We consider the classical Mahonian statistics on the set \(B_n(\Sigma)\) of signed permutations in the hyperoctahedral group \(B_n\) which avoid all patterns in \(\Sigma\), where \(\Sigma\) is a set of patterns of length two. \textit{R. Simion} [Electron. J. Comb. 7, No. 1, Research paper R9, 27 p. (2000; Zbl 0938.05003)] gave the cardinality of \(B_n(\Sigma)\) in the cases where \(\Sigma\) contains either one or two patterns of length two and showed that \(\left|B_n(\Sigma)\right|\) is constant whenever \(\left|\Sigma\right|=1\), whereas in most but not all instances where \(\left|\Sigma\right|=2\), \(\left|B_n(\Sigma)\right|=(n+1)!\). We answer an open question of Simion by providing bijections from \(B_n(\Sigma)\) to \(S_{n+1}\) in these cases where \(\left|B_n(\Sigma)\right|=(n+1)!\). In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on \(B_n(21, \bar{2}\bar{1})\) and by giving the major index on \(D_n(\Sigma)\) for \(\Sigma =\{21, \bar{2}\bar{1}\}\) and \(\Sigma=\{12,21\}\). The main result of this paper is to give the inversion generating functions for \(B_n(\Sigma)\) for almost all sets \(\Sigma\) with \(\left|\Sigma\right|\leq2.\)