Integer sequences | Enumerative combinatorics

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 1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... (sequence in the OEIS). 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). (Wikipedia).

Ordered Bell number
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Related pages

Lowest common ancestor | Fubini's theorem | Guido Fubini | Summation | Eulerian number | Identity matrix | Total order | Bijective proof | Truncated octahedron | Enumerative combinatorics | Factorial | Tree (graph theory) | Weak ordering | Multiplicative partition | Stirling numbers of the second kind | Bell number | Binomial coefficient | Recurrence relation | Permutohedron | Set (mathematics) | Partition of a set | Pascal's triangle | Number theory | Finitary relation | Contour integration | Modular arithmetic