移位U型标准杨表计数公式

Enumeration of shifting U-shaped standard Young tableaux

研究了移位U型半封闭模型标准杨表的计数问题,该模型由U型半封闭模型在西南角的第一个单元格向东北角折叠得到。根据嵌套顺序统计量与标准杨表之间的对应关系,将标准杨表计数问题转化为相应的嵌套单形计算问题,运用多重积分的求解方法,通过积分区域的划分得到标准杨表的数目;再利用组合恒等式、组合求和公式和卡特兰数对计数结果进行化简整理,得到该模型的标准杨表计数公式;最后,针对m=1时的情况,给出了该类型标准杨表在特殊情况下的结论。

This paper studies the enumeration problem of the standard Young tableaux in the shifted U-shaped semi-closed model,which is obtained by folding the first cell of the U-shaped in the southwest corner to the northeast corner.Firstly,the SYT-type chart counting problem is transformed into the corresponding nested monomorphic computational problem based on the correspondence between the nested order statistic and the standard Young tableaux.The number of SYT-type charts is obtained by using the solution method of multiple integrals.Then,the counting results are simplified by using the combination identity,combination summation formula and Catalan number.The SYT-type chart counting formula for the model is obtained.Finally,for the case of m=1,the conclusion of the kind of SYT-type chart in the special case is given.