Rook numbers and the normal ordering problem

TL;DR

This paper establishes a connection between normal ordering coefficients in the Weyl algebra and rook numbers on Ferrers boards, providing explicit formulas and extending results to q-analogues and generalizations.

Contribution

It introduces a novel interpretation of normal order coefficients as rook numbers, offers explicit formulas, and extends the theory to q-analogues and broader rook number generalizations.

Findings

Normal order coefficients correspond to rook numbers on Ferrers boards.

Derived explicit formulas for Weyl binomial coefficients.

Extended results to q-analogues and i-rook number generalizations.

Abstract

For an element $w$ in the Weyl algebra generated by $D$ and $U$ with relation $DU=UD+1$, the normally ordered form is $w=\sum c_{i,j}U^iD^j$. 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 all these results to the $q$-analogue of the Weyl algebra. We discuss further generalizations using $i$-rook numbers.