Let $n$ be a positive integer. A family $A$ of subsets of $\{1,2,\dots,n\}$ is an antichain if for any sets $X,Y\in A$, $X$ is not a subset of $Y$. What is the number of (i) antichains [(ii) non-isomorphic antichains resp.] on $\{1,2,\dots,n\}$ of a fixed size $k$?
For e.g., $n=3$, some antichains of size $k=2$ are
\begin{align*} A_1&=\{\{1\}, \{2,3\}\},\\ A_2&=\{\{2\}, \{1,3\}\},\\ A_3&=\{\{1,2\}, \{2,3\}\}. \end{align*}
$A_1$ and $A_2$ are isomorphic, and both of them are not isomorphic to $A_3$.
An antichain $A$ is isomorphic to another antichain $B$, if there exists a bijection $f$ from $\{1,2,\dots,n\}$ to $\{1,2,\dots,n\}$ such that for any $X \subseteq\{1,2,\dots,n\}$, $X$ is in $A$ if and only in $f(X)$ is in $B$, where $f(X)$ is defined to be $f(X):=\{f(x)\mid x\in X\}$.
Is this a solved question? I am solving a problem that I think is related to this question. Please recommend papers/textbooks on this please. Thank you all.