I have a question which is a refined version of the question asked here. Let $A = \{1,\ldots,n\}$, $S_r = \{a \subset A: |a| = r\}$ for $0 < r \leq n$, $S_r^m = \{(a_1,\ldots,a_m) : a_i \in S_r \}$ for $m>0$, and consider, for $0 < d \leq m$, the set $$ U^n_{r,m,d} = \{ T \in S_r^m : \text{ the union of any $d$ components of $T$ equals $A$}\} \subset S_r^m $$
Note that $U^n_{r,m,1} \subset U^n_{r,m,2} \subset \cdots \subset U^n_{r,m,m} \subset S_r^m$.
I'm interested in computing $\tau(n,r,m,d) \triangleq |U^n_{r,m,d}|$.
My first guess is to use an inclusion-exclusion argument. I'll run through it to illustrate where it gets stuck (as far as I can tell). We try to calculate the cardinality of the complement of $U^n_{r,m,d}$, say $S_r^m = U^n_{r,m,d} \sqcup V^n_{r,m,d}$. For $\ell \in A$ let $ K_{\ell} = \{ (a_1,\ldots,a_m) \in S_r^m : \exists i_1,\ldots,i_d \text{ with } \ell \notin \cup_j a_{i_j} \}$, wherefore $$ V^n_{r,m,d} = \bigcup_{\ell \in A} K_\ell $$ To proceed we are required to be able to express the cardinality of $\bigcap_{j \in J} K_j$, for $J \subset A$. I don't know how to do this. It seems tricky to count: if $(a_1,\ldots,a_m) \in K_1 \cap K_2$, we know that for some $i_1,\ldots,i_d$, $1 \notin \cup_j a_{i_j}$, and for some $k_1,\ldots,k_d$, $2 \notin \cup_j a_{k_j}$, but these tuples may not have any relation to one another in general.
When $m = d$, it simplifies because they must be the same $m$-tuple of indices. Namely we know that $$ \bigcap_{j \in J} K_j = \{(a_1,\ldots,a_m) \in S_r^m : \exists \ell \in J , \ell \notin \cup_i a_i \}$$ The cardinality of this is simply $\binom{n-j}{r}^m$, the result of independently choosing a subset of size $r$ from $A - J$ a total of $m$ times. One concludes that
\begin{align} \tau(n,r,m,m) = | S_r^m| - |V^n_{r,m,d}| &= \binom{n}{r}^m -\sum_{j = 1}^{n} (-1)^{j-1} \binom{n}{j} \binom{n-j}{r}^m \\ &=\sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \binom{n-j}{r}^m \end{align} Note that the upper bound for the sum in the answer to the other problem is (with respect to the variables here) $n - r$ and not $n$ as it is here. If mine is the one which is mistaken please leave a comment.