Bell number

As they show, the number of different combinations in such a system is given by the ordered Bell numbers.

The binomial transform is the shift operator for the Bell numbers.

Because of this correspondence, the number of faces is given by the ordered Bell numbers.

Fifty-two is the 6th Bell number and a decagonal number.

Several asymptotic formulae for the Bell numbers are known.

The number on the left hand side of a given row is the Bell number for that row.

Weak orderings are counted by the ordered Bell numbers.

More generally, the Bell numbers satisfy the following recurrence:

The first few Bell numbers that are primes are:

All of the cumulants of the sequence of Bell numbers are equal to 1.

The corresponding numbers for the permutohedron are respectively the ordered Bell numbers and the factorials.

These numbers are also called the Fubini numbers or ordered Bell numbers.

A fortiori, C never exceeds the nth Bell number.

The ordered Bell numbers appear in the work of , who used them to count certain plane trees with n + 1 totally ordered leaves.

The Bell numbers satisfy this recursion formula:

The ordered Bell numbers may be computed via a summation formula involving binomial coefficients, or by using a recurrence relation.

The exponential generating series for P is , which is the series for the Bell numbers.

He is the eponym of the Bell polynomials and the Bell numbers of combinatorics.

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.

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.

The Touchard polynomials (and thereby the Bell numbers) can be generalized, using the real part of the above integral, to non-integer order:

This number has come to be called the nth Bell number B, after Eric Temple Bell.

Gertsch, Anne; Robert, Alain M.: Some congruences concerning the Bell numbers.

In general, the number of multiplicative partitions of a squarefree number with i prime factors is the ith Bell number, B.

He also worked on linear algebra, matrices, various geometries, topology and Listing numbers, Bell numbers, graphs, the four-color problem, and the nature of continuity.