论文标题
列举标准的几乎设置值的列出陈述
Enumeration of standard barely set-valued tableaux of shifted shapes
论文作者
论文摘要
标准的几乎没有设置的形状$λ$的标准值是填充年轻图$λ$的整数$ 1,2，\ dots，|λ|+1 $，因此整数在每个行和列中都在增加，并且每个单元格都包含一个整数，除了一个包含两个整数的单元格外。计数标准的几乎没有设置值的tableaux与Young晶格中较低间隔的巧合下级期望（CDE）密切相关。使用$ q $ - 综合技术，我们为标准的几乎单位值的tableaux数量提供了一个公式。我们展示了如何用于恢复最初由Reiner，Tenner和Yong猜想的两个公式，并由霍普金斯（Hopkins）证明，用于数量的数量，这些标准几乎设置为特定变化平衡的形状。我们还证明了Reiner，Tenner和Yong在移位形状$（N，N，N-2，N-4，\ DOTS，N-2K+2）$的CDE属性上的猜想。最后，在附录中，我们对$ \ mathsf a; q $ - Q $ analogue提出了一个猜想。对于Young晶格的特定低阶理想的统一分布，对均匀分布的Q $ analogue提出了猜想。
A standard barely set-valued tableau of shape $λ$ is a filling of the Young diagram $λ$ with integers $1,2,\dots,|λ|+1$ such that the integers are increasing in each row and column, and every cell contains one integer except one cell that contains two integers. Counting standard barely set-valued tableaux is closely related to the coincidental down-degree expectations (CDE) of lower intervals in Young's lattice. Using $q$-integral techniques we give a formula for the number of standard barely set-valued tableaux of arbitrary shifted shape. We show how it can be used to recover two formulas, originally conjectured by Reiner, Tenner and Yong, and proved by Hopkins, for numbers of standard barely set valued tableaux of particular shifted-balanced shapes. We also prove a conjecture of Reiner, Tenner and Yong on the CDE property of the shifted shape $(n,n-2,n-4,\dots,n-2k+2)$. Finally, in the Appendix we raise a conjecture on an $\mathsf a;q$-analogue of the down-degree expectation with respect to the uniform distribution for a specific class of lower order ideals of Young's lattice.
