A recursion formula for finite partition lattices

Abstract

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 Mobius function for partition lattices different from those in [2], [3] and [5]. Let S be a finite nonempty set. Two partitions oand 7r of S satisfy o Z by setting k(o)= HTB h(IBI) for oEL(S) (IBI =the cardinality of the block B). It is then possible to define another function H from the nonnegative integers into Z by setting H(O) = 1 and H(n) = ZaeLn k(of) for n > 1, where Ln is the lattice of partitions on a set with n elements.