Q-Counting rook configurations and a formula of Frobenius
Journal of Combinatorial Theory Series A
European Journal of Combinatorics - Special issue on combinatorics and representation theory
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We study a q-analog Q"r(n,q) of the partition algebra P"r(n). The algebra Q"r(n,q) arises as the centralizer algebra of the finite general linear group GL"n(F"q) acting on a vector space IR"q^r coming from r-iterations of Harish-Chandra restriction and induction. For n=2r, we show that Q"r(n,q) has the same semisimple matrix structure as P"r(n). We compute the dimension d"n","r(q)=dim(IR"q^r) to be a q-polynomial that specializes as d"n","r(1)=n^r and d"n","r(0)=B(r), the rth Bell number. Our method is to write d"n","r(q) as a sum over integer sequences which are q-weighted by inverse major index. We then find a basis of IR"q^r indexed by n-restricted q-set partitions of {1,...,r} and show that there are d"n","r(q) of these.