Enumeration of standard barely set-valued tableaux of shifted shapes

A standard barely set-valued tableau of shape λ is a filling the Young diagram with integers 1,2,…,|λ|+1 such that are increasing in each row and column, every cell contains one integer except two integers. Counting tableaux closely related to coincidental down-degree expectations (CDE) lower intervals Young’s lattice. Using q-integral techniques we give formula for number arbitrary shifted shape. We show how it can be used recover formulas, originally conjectured by Reiner, Tenner Yong, proved Hopkins, numbers particular shifted-balanced shapes. also prove conjecture Yong on CDE property (n,n−2,n−4,…,n−2k+2). Finally, appendix raise an a;q-analogue expectation respect uniform distribution specific class