After this attempt I’ll try to advance the problem of counting pos seriously, in a new post: pos(n).
Positive integers are called natural numbers. When also integer is included then they are called non-negative integers.
I’ll be concerned with the finite sets only. A partial order is abbreviated as po. Pos are in a bijective (1-1) correspondence with the -topologies. We want to count the number of pos in arbitrarily fixed (finite) set, or–what is essentially the same–the -topologies (in the same set).
Let be the number of coverings of an -element set by sets , i.e. (it’s a nice mood improver, i.e. a simple exercise that is easier than it looks). The sets can be empty.
The following formula:
holds for arbitrary non-negative ; here the exponentiation conventions honors equality .
Pos or -topologies. Signature.
A sequences of non-negative integers is called a signatures if there exists a zero element ; and .
Next, we associate a unique signature with every finite poset. If two such sets are isomorphic then they will have the same signature.
Consider a finite poset (partially ordered set) (thus ). Then let be the set of the minimal elements of ; and where and .
Let’s continue these definitions: , where: