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Let $\mathsf Y_{N,r}$ be the set of all Young diagrams with $N$ boxes and no more than $r$ rows. Let $d_y$ be the degree of the irreducible representation of GL(n) corresponding to the Young diagram $y$.

Question: Is there a closed formula for $s_{N,r,n}:=\sum_{y \in \mathsf Y_{N,r}} \, d_y$, with $2\le r\le n$? Or a good lower bound in the large N limit?

(I am looking for the generalization of the formula $s_{N,2,2} = (N/2+1)^2$, $N$ even, to general $r$ and $n$)

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