The number of partially ordered sets

Abstract

An unsolved problem in combinatorial analysis asks for G ∗(n) , the number of different partial orderings which may be defined on a finite set containing n elements [2]. In the present paper we shall give a partial solution to this problem by interpreting a partial order relation as a non-singular idempotent Boolean relation matrix. The main results of this paper are as follows: (i) Enumerating the partial order relations which may be defined on a finite set containing n elements is equivalent to enumerating E( n), the set of all n × n non-singular idempotent Boolean relation matrices. (ii) The set E( n) and a special subset E( n, r) are defined. (iii) The number of interest is |E(n)|= ∑ r=0 n(n−1) 2 |E(n,r)| (iv) Formulas for | E( n, r)| ( r = 0, 1, 2, 3, ( n( m−1)/2) −2, ( n( n−1)/2)−1, and n( n−1)/2) are given.