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 = Sigma 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 ci, 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. (c) 2005 Elsevier Inc. All rights reserved.