Systematic vertex information about finite Young's lattice

Question

Suppose that I have Young's lattice $Y_{\mu}$ (finite sublattice for everything included in $\mu$). See the red area in the linked picture I found online where $\mu=[22]$ and note that the arrows indicating inclusion are missing (should be going from left to right).

$Y_{[22]}$

My question is simple: is there a way of systematically counting the number of 'incoming' edges for each vertex in the direction of inclusion? In the depicted case I would like to count the edges of every vertex (partition) inside the red area coming from left. Note that I'm not asking how to do it for a given partition -- that is easy. I'm interested in a systematic approach, for example, by a generating function. Actually, it could be a fine-grained rank-generating function which in this case is given by the $q$-binomial $\binom{4}{2}_q=1 + q + 2 q^2 + q^3 + q^4$. The answer to my question for $Y_{[22]}$ would be $$\{0\},\{1\},\{\{1\},\{1\}\},\{2\},\{1\}$$ Zero incoming edge for $\varnothing$, one incoming edge for $\mu=[1]$ etc...