StirlingS1

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

(((-1))^(n-m))(Subscript[S, n]^((m))) is the number of ways of partitioning a set of n elements which contain exactly m permutation disjoint cycles.

Specific values

Specialized values

For fixed n

For fixed m

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[S, n]^((m))) is a nonanalytical function which is defined only for nonnegative integers n,  m.

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

No periodicity

Series representations

Generalized power series

Asymptotic series expansions

Residue representations

Integral representations

Contour integral representations

Limit representations

Generating functions

Identities

Recurrence identities

Functional identities

Identities involving determinants

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Summation

Finite summation

Not involving StirlingS2

Involving StirlingS2

Infinite summation

Representations through equivalent functions

With related functions

Theorems

Disjoint cycles in permutations

(Subscript[S, n]^((m))) is the number of permutations of m symbols that have n disjoint cycles.

Converting derivatives to finite differences

With the finite difference operator (Subscript[Δ, x,h]^k) defined by (Subscript[Δ, x,h]^k)f(x)=(Subscript[Δ, x,h]^(k-1))((Subscript[Δ, x,h])f(x)), (Subscript[Δ, x,h])f(x)=f(x+h)-f(x) the kth derivative of a function f(x) can be expressed as

(((∂^k)f(x))/(∂(x^k)))= (∑_(j=0)^∞)((h^(-k))/(Subscript[(k+1), j]))(Subscript[S, j+k]^((k))) (Subscript[Δ, x,h]^(k+j))f(x).

History

– J. Stirling (1730, 1749)

Copyright