In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
and
enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infin ...
, the ordered Bell numbers or Fubini numbers count the number of
weak ordering
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered ...
s on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of ''n'' elements (orderings of the elements into a sequence allowing
ties TIES may refer to:
* TIES, Teacher Institute for Evolutionary Science
* TIES, The Interactive Encyclopedia System
* TIES, Time Independent Escape Sequence
* Theoretical Issues in Ergonomics Science
The ''Theoretical Issues in Ergonomics Science' ...
, such as might arise as the outcome of a
horse race).
[. Because of this application, de Koninck calls these numbers "horse numbers", but this name does not appear to be in widespread use.] Starting from ''n'' = 0, these numbers are
:1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... .
The ordered Bell numbers may be computed via a summation formula involving
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s, or by using a
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
. Along with the weak orderings, they count several other types of combinatorial objects that have a
bijective correspondence to the weak orderings, such