Note on the number of antichains in generalizations of the Boolean lattice

We give a short and self-contained argument that shows that, for any positive integers $t$ and $n$ with $t=O\left(\frac{n}{\log n}\right)$, the number $\alpha ([t{]}^n)$ of antichains of the poset $[t{]}^n$ is at most $${\exp }_2(1+O(\left(\frac{t{\log }^{3}n}{n}{)}^{1\mathrm{/}2}\right))N(t,n),$$ where $N(t,n)$ is the size of a largest level of $[t{]}^n$. This, in particular, says that if $t\ll n\mathrm{/}{\log }^{3}n$ as $n\to \mathrm{\infty }$, then $\log \alpha ([t{]}^n)=(1+o(1))N(t,n)$, giving a (partially) positive answer to a question of Moshkovitz and Shapira for $t,n$ in this range. Particularly for $t=3$, we prove a better upper bound: $$\log \alpha ([3{]}^n)\le (1+4\log 3\mathrm{/}n)N(3,n),$$ which is the best known upper bound on the number of antichains of $[3{]}^n$.