Stirling numbers

Several different notations for the Stirling numbers are in use.

Various notations have been used for Stirling numbers of the second kind.

Hence the name "Stirling numbers of the second kind."

If denotes Stirling numbers of the second kind then one has:

Most identities on this page are stated for unsigned Stirling numbers.

Note that again generates the unsigned Stirling numbers of the first kind, but in reverse order.

The Stirling numbers of the second kind are given by the explicit formula:

This article is devoted to specifics of Stirling numbers of the first kind.

The original definition of Stirling numbers of the first kind was algebraic.

The Lah numbers are sometimes called Stirling numbers of the third kind.

Many relations for the Stirling numbers shadow similar relations on the binomial coefficients.

Furthermore, the number of ways to partition a set into exactly k blocks we use the Stirling numbers .

Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials.

In mathematics, Stirling numbers arise in a variety of analytic and combinatorics problems.

Identities linking the two kinds appear in the article on Stirling numbers in general.)

We prove the recurrence relation using the definition of Stirling numbers in terms of rising factorials.

Observe (both by definition and by the reduction formula), that , the familiar Stirling numbers of the second kind.

Lah numbers are related to Stirling numbers.

The Stirling numbers of the first and second kinds can be considered to be inverses of one another:

The Stirling numbers of the second kind can represent the total number of rhyme schemes for a poem of n lines.

Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions.

This EGF yields the formula for the Stirling numbers of the second kind:

Its coefficients are expressible in terms of Stirling numbers of the first kind, by definition of the latter:

The Lagrange interpolation formula and Stirling numbers, Proc.

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations.