How many blocks can be made of a subset of S in a Steiner system Assume a Steiner system with parameters $t, k, n$, i.e., $S(t,k,n)$. By definition, the system is an $n$-element set $S$ together with a set of $k$-element subsets of $S$ (called blocks) such that each $t$-element subset of $S$ is contained in exactly one block.
Now consider a Steiner system $S(t,k,n)$ and a subset $X \subset S$ with cardinality $|X| = c > k$. The question I am interested in is:

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*How many blocks of the system consist exclusively of elements of $X$?

An expression given at Wikipedia quantifies how many blocks include a $c$-element subset of S and holds for values of $c \leq t$.
I am wondering whether there is an expression for all values of $c \in \{t+1,\dots,k\}$ for the question stated above. I would think that the answer depends on the construction method. Are there constructions for which one can give such an expression?
 A: If we let $\mathcal{D}$ be an $S(t,k,n)$, in other words, a $t$-$(n,k,1)$ design, then the complementary design $\mathcal{D}^{c}$, obtained by replacing each block with its complement, is also a $t$-design (though not a Steiner system), in other words it is a $t$-$(n,n-k,\lambda^{\prime})$ for some $\lambda^{\prime}$.
The Wikipedia result you give says that if $\mathcal{D}$ is an $S(t,k,n)$ then for each $0 \leq i \leq t$, we have that there are
$$\lambda_{i} = {n-i \choose t-i}/{k-i \choose t-i}$$
blocks of $\mathcal{D}$ containing any fixed $i$-subset of $\{1,\ldots, n\}$.
This is however a special case of a result that holds for any $t$-$(n,k,\lambda)$ design that gives this value as
$$\lambda_{i} = \lambda{n-i \choose t-i}/{k-i \choose t-i}$$
If we want to know how many blocks of $\mathcal{D}$ are contained in some fixed $j$-set $X$, this number will be constant over all possible $j$-sets as long as $n-t \leq j \leq n$.
To see this, we should realize that this is equivalent to asking how many blocks of $\mathcal{D}^{c}$ contain $X^{c}$ (or rather consider $X^{c}$ to be an arbitrary $(n-j)$-subset).
In this case, modifying the formula above with the parameters of $\mathcal{D}^{c}$, we have
$$\lambda^{\prime}_{j} = 
\lambda^{\prime}\frac{{n-(n-j) \choose t-(n-j)}}{{n-k-(n-j) \choose t-(n-j)}} = 
\lambda^{\prime}\frac{{j \choose j-(n-t)}}{{j-k \choose j-(n-t)}} = 
\lambda^{\prime}\frac{{j \choose n-t}}{{j-k \choose j-(n-t)}}$$
where $\lambda^{\prime}$ is given by
$$\lambda^{\prime} = \sum_{i=0}^{t} (-1)^{i}{t \choose i}\lambda_{i}.$$
(See Godsil, Combinatorial Design Theory Lemma 7.3.1)
Godsil actually gives an easier formula:
$$\lambda_{j}^{\prime} = \sum_{i=0}^{j} (-1)^{i}{j \choose i}\lambda_{i}$$
(in both of these formulas the $\lambda_{i}$ are given by the earlier formula $\lambda_{i} = {n-i \choose t-i}/{k-i \choose t-i}$).
Note: if $j < k$ then $X$ cannot contain any blocks of $\mathcal{D}$; and if $k \leq j < n-t$ then the number of blocks contained in $X$ will depend on the choice of the set $X$ and not strictly on $|X|$. (For example if $j=k$ then the number of blocks contained in $X$ will be $1$ if $X$ is a block of $\mathcal{D}$, and $0$ otherwise.)
