I encountered the following combinatorics problem in my research, and I'd like to know if there is a reference or an easy solution for such a problem.
Given a partially ordered set $\mathscr P$, an antichain is a subset of $\mathscr P$ such that no two elements can be compared.
Fix positive integers $n$ and $k$. Define $\mathscr A_{n,k}$ to be the set of all size-$k$ antichains taken from the poset of subsets of $[n] = \{1, \cdots n \}$. Define $A_{n,k} = \mathscr A_{n,k} / S_n$ where the permutation group $S_n$ acts on $\mathscr A_{n,k}$ by extending the natural action of $S_n$ on $[n]$; for $\sigma \in S_n$ and $\mathscr U \in \mathscr A_{n,k}$, $\sigma( \mathscr U) := \{\sigma(U) | U \in \mathscr U \}$. Alternatively, \begin{align*} & \mathscr A_{n,k} := \left\{ \{ U_1, \dots, U_k \} | \emptyset \neq U_i \subseteq [n], \forall i \neq j, U_i \not\subseteq U_j \right\} \\ & A_{n,k} := \mathscr A_{n,k} / S_n \end{align*}
Problem. Enumerate all elements of $A_n = \bigcup_{k=1}^\infty A_{n,k}$.
The problem is in data science context; I'd like to have an algorithm that computes, say, $A_{10}$ that doesn't take too long. But I'll be computing it only once, so it's fine if the computation takes a whole week (but not a whole year!). Naive enumeration and checking for all overlaps will take quite long; for example, the set of all size-5 antichains of $[10]$ has size $(2^{10})^5 \approx 10^{15}$.
Remark. One way to understand $\mathscr A_{n,k}$ is to see it as the collection of all hypergraphs on $[n]$ such that one hyperedge never includes another; it is of completely opposite nature to abstract simplicial complexes. Also, following Brendan McKay's observation, I edited the question and used the antichain terminology.
