$q$-Partition Algebra Combinatorics

We compute the dimension $d_{n,r}(q) = \dim(\IR_q^r)$ of the defining module $\IR_q^r$ for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on $\GL_n(\FF_q)$. This dimension is a polynomial in $q$ that specializes as $d_{n,r}(1) = n^r$ and $d_{n,r}(0) = B(r)$, the $r$th Bell number. We compute $d_{n,r}(q)$ in two ways. The first is purely combinatorial. We show that $d_{n,r}(q) = \sum_\lambda f^\lambda(q) m_r^\lambda$, where $f^\lambda(q)$ is the $q$-hook number and $m_r^\lambda$ is the number of $r$-vacillating tableaux. Using a Schensted bijection, we write this as a sum over integer sequences which, when $q$-counted by inverse major index, gives $d_{n,r}(q)$. The second way is algebraic. We find a basis of $\IR_q^r$ that is indexed by $n$-restricted $q$-set partitions of $\{1,..., r\}$, and we show that there are $d_{n,r}(q)$ of these.