StirlingS2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

(Subscript[𝒮, n]^((m))) is the number of ways of partitioning a set of n elements into m nonempty subsets.

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 StirlingS1

Involving StirlingS1

Infinite summation

Operations

Limit operation

Representations through equivalent functions

With related functions

Theorems

Number of surjective functions

n! (Subscript[𝒮, n]^((m))) is the number of surjective functions from a set with m elements to a set with n elements.

Derivative of composition with exponential function

(((∂^n)f((e^z)))/(∂(z^n)))==(∑_(k=1)^n)(e^(k z)) (Subscript[𝒮, n]^((k))) Derivative[k][f]((e^z))/;n∈(SuperPlus[ℕ]) .

Converting finite differences to derivatives

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+1)-f(x) the kth finite difference of a function f(x) can be expressed through derivatives as

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

History

–J. Stirling (1730)

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