# Ordered Bell number

In number theory and enumerative combinatorics, the **ordered Bell numbers** or **Fubini numbers** count the number of weak orderings on a set of *n* elements (orderings of the elements into a sequence allowing ties, such as might arise as the outcome of a horse race).[1] Starting from *n* = 0, these numbers are

The ordered Bell numbers may be computed via a summation formula involving binomial coefficients, or by using a recurrence relation. Along with the weak orderings, they count several other types of combinatorial objects that have a bijective correspondence to the weak orderings, such as the ordered multiplicative partitions of a squarefree number[2] or the faces of all dimensions of a permutohedron[3] (e.g. the sum of faces of all dimensions in the truncated octahedron is 1 + 14 + 36 + 24 = 75[4]).