AI 中文总结
研究$(p,q)$-变形广义Weyl代数中正规排序产生的组合结构，通过引入$(p,q)$-变形$s$-车数，给出$(p,q)$-广义Stirling数在阶梯棋盘上车放置的组合解释。
AI 中文摘要
本文研究了由变量$X, Y$和$Z_p$生成的$(p,q)$-变形广义Weyl代数所产生的组合结构，该代数满足$(p,q)$-交换关系$XY-qYX=h Y^sZ_{p}$、$XZ_p=pZ_pX$和$Z_pY=pYZ_p$，其中$s\in \mathbb{N}_0$。我们的主要目标是利用这些关系定义的正规排序过程，发展一种新的$(p,q)$-变形车理论模型。具体地，我们引入了一个新的$(p,q)$-变形$s$-车数框架，该框架源于此正规排序过程。利用这些组合模型，我们通过阶梯棋盘上的车放置，为相关的$(p,q)$-广义Stirling数提供了显式的组合解释。我们的结果将文献中若干经典和近期的公式推广到一般的$p\neq 1$情形。
英文摘要
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 $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, 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 $p\neq 1$ setting.
Comments23 pages
