Descent sets of cyclic permutations in types B and D

Abstract

Elizalde constructed a bijection $\phi$ from the cyclic permutations $\pi\in S_{n+1}$ to the symmetric group $S_n$ satisfying $\operatorname{Des}(\pi)\cap \{1,2,\ldots,n-1\}=\operatorname{Des}(\phi(\pi))$. We give a corresponding result on the signed symmetric group $B_n$ by constructing a function $\Phi$ from the cyclic signed permutations $\pi\in B_{n+1}$ to $B_n$ satisfying $\operatorname{Des}(\pi)\cap \{0,1,\ldots,n-1\}=\operatorname{Des}(\Phi(\pi))$. Moreover, letting $D_{n+1}\subseteq B_{n+1}$ be the subgroup consisting of signed permutations with an even number of sign changes, we show that the restriction of $\Phi$ to the cyclic signed permutations in $D_{n+1}$ or its complement is a bijection. Our function $\Phi$ reduces to Elizalde's original bijection $\phi$ under the natural identification of the symmetric groups as subgroups of the signed symmetric groups. One application of our results is asymptotic normality of the descent and flag major index statistics on the cyclic signed permutations in $B_{n}$ and $D_n$.