Fubini Number Calculator

Named after Guido Fubini who studied ordered partitions. Also called 'ordered Bell numbers' because Bell numbers count unordered partitions and Fubini = Bell × ordering. The connection: a(n) = Σ B_{n,k}·k! where B_{n,k} is a partial Bell number.

A total order allowing ties. For {a,b,c}: strict orderings like a<b<c (6 total), orderings with one tie like a<b=c (6 total), and the single 3-way tie a=b=c. Total: 6+6+1=13=a(3).

a(n) ~ n!/(2(ln 2)^{n+1}). This is faster than n! by a factor of 1/(2(ln 2)^{n+1}), which grows superexponentially. a(10) = 102,247,563. They grow roughly n!/0.693^n.