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 ${\displaystyle n}$ elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race). Starting from ${\displaystyle n=0}$, 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 or the faces of all dimensions of a permutohedron.