degree $n$, let $\beta(v)$ be the coefficient of $v$ in $\Psi(B_{n+1})$. Our primary objective is to understand $\beta$.

1.  We know that $\Psi(B_{n})$ is a polynomial whose coefficients are positive integers. What do these numbers count?

    The [**cd**]{}-index of $B_n$ is a refined enumeration of André permutations . Similarly, it is also a refined enumeration of simsun permutations, first defined by Simion and Sundaram . These permutations seem ad hoc and it is not clear how to use them to answer some of the questions asked below. More recently, there has been an interpretation involving the peak statistic of permutations, which may prove more useful.

2.  What can be said about the equalities satisfied by the values of $\beta$ ?

    It is known that $\beta(v) = \beta(v^*)$, where $v^*$ is the monomial $v$ written in reverse order. For small ranks, this can be seen from the data in Figure \[f:bl\]. Hence one may ask: Are there [**cd**]{}-monomials $u$ and $v$ such that $u \not = v^*$, but yet $\beta(u) = \beta(v)$? Lemma \[l:identities\] provides a partial answer to this question. Lemma \[l:switch\] gives a slightly more general answer that involves a different product $\centerdot$ on $k \langle {{\bf c}},{{\bf d}}\rangle$. 