
Naiomi T. Cameron

Kendra Killpatrick
Keywords:
Signed permutations, Pattern avoiding permutations, Inversion statistic, Major index, Generating function
Abstract
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. In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\leftB_n(\Sigma)\right$ is constant whenever $\left\Sigma\right=1$, whereas in most but not all instances where $\left\Sigma\right=2$, $\leftB_n(\Sigma)\right=(n+1)!$. We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\leftB_n(\Sigma)\right=(n+1)!$. 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 $\left\Sigma\right\leq2.$