Number of pos (just an attempt)

Introduction

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   0   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 T_0-topologies. We want to count the number of pos in arbitrarily fixed (finite) set, or–what is essentially the same–the T_0-topologies (in the same set).

Covers

Let   C(m\ n)   be the number of coverings   \gamma : V\rightarrow 2^W   of an n-element set   W   by   m   sets   \gamma(v),   i.e.   \cup\gamma\ :=\ \cup_{v\in V}\ \gamma(v)\ =\ W   (it’s a nice mood improver, i.e. a simple exercise that is easier than it looks). The sets   \gamma(v)   can be empty.

The following formula:

            C(m\ n)\ =\ (2^m-1)^n

holds for arbitrary non-negative   m\ n;   here the exponentiation conventions honors equality   C(0\ 0) = 1.

Pos or T_0-topologies. Signature.

A sequences of non-negative integers   n_0\ n_1\ \ldots   is called a signatures   \Leftarrow:\Rightarrow   if there exists a zero element   n_k=0;   and   \forall_{i\ j}\ \ \left(\ \left(\left(i<j\right)\ \&\ \left(n_i=0\right)\right)\ \Rightarrow\ \left(n_j=0\right)\ \right).

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)   \mathbf X := (X\ \preceq)   (thus   \preceq\subseteq X\times X).   Then let   M(\mathbf X)   be the set of the minimal elements of   \mathbf X;   and   \mathbf N(\mathbf X) := (N(\mathbf X)\ \preceq|N(\mathbf X)   where   N(\mathbf X) := X\setminus M(\mathbf X)   and   \preceq | N(\mathbf X)\ :=\,\ \preceq\ \cap\ \left(N\left(\mathbf X\right) \times N\left(\mathbf X\right)\right).

Let’s continue these definitions:   \mathbf X_n := (X_n\ \preceq_n),   where:

  • X_0 := X\qquad \preceq_0\ :=\ \preceq \qquad M_0 := M(\mathbf X);
  • X_{n+1} := N(\mathbf X_n)\qquad \preceq_{n+1}\ :=\ \preceq | X_{n+1}\qquad M_{n+1} := M(X_{n+1}).

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