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). 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 or the faces of all dimensions of a permutohedron (e.g. the sum of faces of all dimensions in the truncated octahedron is 1 + 14 + 36 + 24 = 75). History. The ordered Bell numbers appear in the work of , who used them to count certain plane trees with "n" + 1 totally ordered leaves. In the trees considered by Cayley, each root-to-leaf path has the same length, and the number of nodes at distance "i" from the root must be strictly smaller than the number of nodes at  distance "i" + 1, until reaching the leaves. In such a tree, there are "n" pairs of adjacent leaves, that may be weakly ordered by the height of their lowest common ancestor; this weak ordering determines the tree. call the trees of this type "Cayley trees", and they call the sequences that may be used to label their gaps (sequences of "n" positive integers that include at least one copy of each positive integer between one and the maximum value in the sequence) "Cayley permutations". These numbers were called Fubini numbers by Louis Comtet, because they count the number of  different ways to rearrange the ordering of sums or integrals in Fubini's theorem, which in turn is named after Guido Fubini. For instance, for a bivariate integral, Fubini's theorem states that where these three formulations correspond to the three weak orderings on two elements. In general, in a multivariate integral, the ordering in which the variables may be grouped into a sequence of nested integrals forms a weak ordering. The Bell numbers, named after Eric Temple Bell, count the number of partitions of a set, and the weak orderings that are counted by the ordered Bell numbers may  be interpreted as a partition together with a total order on the sets in the partition. Formula. The "n"th ordered Bell number may be given by a summation formula involving the Stirling numbers of the second kind, which count the number of partitions of an "n"-element set into "k" nonempty subsets, expanded out into a double summation involving binomial coefficients (using a formula expressing Stirling numbers as a sum of binomial coefficients), or given by an infinite series: An alternative summation formula expresses the ordered Bell numbers in terms of the Eulerian numbers, which count the number of permutations of  n items with "k" + 1 runs of increasing items: where "A" is the "n"th Eulerian polynomial. The exponential generating function of the ordered Bell numbers is This can be expressed equivalently as the fact that the ordered Bell numbers are the numbers in the first column of the infinite matrix (2"I" − "P"), where "I" is the identity matrix and "P" is an infinite matrix form of Pascal's triangle. Based on a contour integration of this generating function, the ordered Bell numbers can be expressed by the infinite sum and approximated as Because log 2 is less than  one, the form of this approximation shows that the ordered Bell numbers exceed the corresponding factorials by an exponential factor. The asymptotic convergence of this approximation may be expressed as Recurrence and modular periodicity. As well as the formulae above, the ordered Bell numbers may be calculated by the recurrence relation The intuitive meaning of this formula is that a weak ordering on "n" items may be broken down into a choice of some nonempty set of "i" items that go into the first equivalence class of the ordering, together with a smaller weak ordering on the remaining "n" −  "i" items. As a base case for the recurrence, "a"(0) = 1 (there is one weak ordering on zero items). Based on this recurrence, these numbers can be shown to obey certain periodic patterns in modular arithmetic: for sufficiently large "n", Several additional modular identities are given by and . Additional applications. As has already been mentioned, the ordered Bell numbers count weak orderings, permutohedron faces, Cayley trees, Cayley permutations, ordered multiplicative partitions of squarefree numbers, and equivalent formulae in Fubini's theorem. Weak orderings in turn have many other applications. For instance, in horse racing, photo finishes have eliminated most  but not all ties, called in this context dead heats, and the outcome of a race that may contain ties (including all the horses, not just the first three finishers) may be described using a weak ordering. For this reason, the ordered Bell numbers count the possible number of outcomes of a horse race, or the possible outcomes of a multi-candidate election. In contrast, when items are ordered or ranked in a way that does not allow ties (such as occurs with the ordering of cards in a deck of cards, or batting orders among baseball players), the number  of orderings for "n" items is a factorial number "n"!, which is significantly smaller than the corresponding ordered Bell number.