# Enumeration of standard barely set-valued tableaux of shifted shapes

### (28 pages)

**Abstract.**
A standard barely set-valued tableau of shape λ is a filling
of the Young diagram λ with integers 1, 2, …, |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, …, n−2k+2). Finally, in
the appendix we raise a conjecture on an *a*;*q*-analogue of
the down-degree expectation with respect to the uniform distribution for
a specific class of lower intervals in Young's lattice.

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