Rook numbers and the normal ordering problemAnna VarvakBrandeis University
March 12,

ABSTRACT


Let $w$ be an element in the Weyl algebra (generated by two elements $D$ and $U$, with a commutation relation $DUUD=1$.) 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 Theoremthat the generating function of rook numbers on a Ferrers board, expressed in terms of falling factorials $x(x1)...(xk)$, 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 $DUqUD=1$). 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

