Combinatorial aspects of boson algebra

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 $w=\sum c_{i,j}U^iD^j$. 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.

https://arxiv.org/pdf/quant-ph/0501155.pdf

Boson normal ordering via substitutions and Sheffer-type polynomials

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

/pdf/on-a-new-family-of-generalized-stirling-and-bell-numbers-315bsgcz90.pdf

On a new family of generalized Stirling and Bell numbers

It is shown that the various types of normal terms arising from the normal ordering of a disordered product of fermion operators can be enumerated with the aid of a well-known combinatorial problem, called the problem of the rooks

A. M. Navon

Papers

Combinatorics and fermion algebra