A recursion formula for finite partition lattices

Abstract

A partition on a nonempty set 5 is a collection of disjoint nonempty subsets, called blocks, whose union is S. The recursion formula 2Z.1 (n-1\ (1) B» = T,(.)Bi (n^l,B0=l) for the number of partitions on a finite set with n elements is well known (cf. Rota [4]). We obtain equation (2) which generalizes (1) and which yields a derivation of the Möbius function for partition lattices different from those in [2], [3] and [5]. Let S be a finite nonempty set. Two partitions a and it of 5 satisfy a^ir if every block of <r is contained in a block of w. ^ is a partial ordering of the collection L(S) of partitions of S and (L(S), i£) is a lattice (cf. Birkhoff [l]). If A is a function from the natural numbers into the integers Z, it is possible to define a function k: L(S)-*Z by