Young's lattice $Y$ is a graded poset and a distributive lattice whose elements are all the partitions of $n$ for $n \in \mathbb{N}$ with the poset relation coming from inclusion of Young diagrams. Here is wikipedia's picture with the smallest few elements that shows only the adjacent order relations (i.e., one needs transitive closure to get the full picture):

Fix integers $0 < m \leq n$ and let $Y(m,n)$ denote the sub-poset of $Y$ "between" the partitions of $m$ and $n$, both inclusive. So, $Y(3,6)$ is the sub-poset generated by partitions of $3, 4, 5$ and $6$, but nothing else. Since each poset corresponds to a simplicial *order complex* whose $d$-simplices are precisely chains of length $d+1$, each $Y(m,n)$ may be viewed as a simplicial complex. Here's my question:

For each dimension $d$, what is known about the $d$-th Betti number of $Y(m,n)$ as a function of $m$ and $n$?

If there is no simple formula, I'd be happy with asymptotics as $n$ is sent to $\infty$ for each fixed $m$. Note that for $m = 1$ the whole thing is contractible to the unique minimal element, but things (appear to) get more complicated for larger $m$.