# Rook numbers and the normal ordering problem

## ABSTRACT

Let $w$ be an element in the Weyl algebra (generated by two elements $D$ and $U$, with a commutation relation $DU-UD=1$.) The normal ordering problem is to find the normal order coefficients $c_{i,j}$ in the sum $w = \sum c_{i,j} U^i D^j$, where in each term the $D$'s are to the right of the $U$'s.

I'll present my proof that, for $w$ a word in the letters $D$, $U$, the normal order coefficients $c_{i,j}$ are rook numbers on the Ferrers board outlined by $w$.

This provides a new proof of the Rook Factorization Theorem---that the generating function of rook numbers on a Ferrers board, expressed in terms of falling factorials $x(x-1)...(x-k)$, completely factors into linear components.

The Rook Factorization Theorem, in turn, provides an easy algorithm to compute the normal order coefficients of $w$.

I'll discuss a generalization to the normal ordering problem with elements in $q$-Weyl algebra (commutation relation $DU-qUD=1$).

Speaker's Contact Info: anka(at-sign)brandeis.edu