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

TL;DR

This paper determines the number of posets with up to 7 elements and the total number of posets with 10 elements without computer assistance, including Hasse diagrams and formulas for specific poset classes.

Contribution

It provides a manual enumeration of posets for small sizes and formulas for counting connected posets, expanding combinatorial understanding without computational tools.

Findings

Number of posets with 7 elements enumerated

Total number of posets with 10 elements computed

Formulas for counting certain connected posets

Abstract

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 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 $|P(10)|$ 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 $|P(n)|$ with $1\le n\le 8$ by a different method.