Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. II: Interpretation in terms of rook placements

In this paper, we investigate the combinatorial structure arising from the $(p,q)$-deformed generalized Weyl algebra generated by variables $X,Y$, and $Z_p$, satisfying the $(p,q)$-commutation relations $XY-qYX=h{Y}^s{Z}_p,X{Z}_p=p{Z}_{p}X$, and $Z_{p}Y=pY{Z}_p$, where $s\in {\mathbb{N}}_0$. Our primary objective is to use the normal ordering process defined by these relations to develop a novel model of $(p,q)$-deformed rook theory. Specifically, we introduce a new framework of $(p,q)$-deformed $s$-rook numbers derived from this normal ordering process. Utilizing these combinatorial models, we provide explicit combinatorial interpretations for the associated $(p,q)$-generalized Stirling numbers via rook placements on staircase boards. Our results extend several classical and recent formulations in the literature to the general $p\mathrm{\ne }1$ setting.