Rook numbers and the normal ordering problem

For an element w in the Weyl algebra generated by D and U with relation DU = UD + 1 , the normally ordered form is w = ∑ c i , j U i D j . We demonstrate that the normal order coefficients c i , j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c i , j . We calculate the Weyl binomial coefficients: normal order coefficients of the element ( D + U ) n in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.