Hasse diagrams of posets with up to 7 elements, and the number of posets with 10 elements, without the use of computer programs

Let $P(n)$ be the set of all posets with $n$ elements. Let $P^{(j)}(n)$, $1\le j\le 2^n,$ be the number of all posets with $n$ elements possessing exactly $j$ antichains. We have determined the numbers $P^{(j)}(7),$ $1\leq j\leq 128$, and using a result of M.~Ern'e [Ern'e, M., On the cardinalities of finite topologies and the number of antichains in partially ordered sets, Discrete Mathematics 35 (1981), 119-133.], we compute $\mathrm{\mid }P(10)\mathrm{\mid }$ without the aid of any computer program. We include the Hasse diagrams of all the non-isomorphic posets of $P(7)$. We also present formulas for the number of connected posets of certain forms, and use them to compute $\mathrm{\mid }P(n)\mathrm{\mid }$ with $1\le n\le 8$ by a different method.