Enumeration of standard barely set-valued tableaux of shifted shapes

A standard barely set-valued tableau of shape $\lambda$ is a filling of the Young diagram $\lambda$ with integers $1,2,\dots,|\lambda|+1$ 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.