q-Partition algebra combinatorics

We study a q-analog Qr(n,q) of the partition algebra Pr(n). The algebra Qr(n,q) arises as the centralizer algebra of the finite general linear group GLn(Fq) acting on a vector space coming from r-iterations of Harish–Chandra restriction and induction. For n⩾2r, we show that Qr(n,q) has the same semisimple matrix structure as Pr(n). We compute the dimension to be a q-polynomial that specializes as dn,r(1)=nr and dn,r(0)=B(r), the rth Bell number. Our method is to write dn,r(q) as a sum over integer sequences which are q-weighted by inverse major index. We then find a basis of indexed by n-restricted q-set partitions of {1,…,r} and show that there are dn,r(q) of these.