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

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