Evaluate a Stirling number of the second kind:

Evaluate multiple Stirling numbers:

StirlingS2 threads element-wise over lists:

TraditionalForm formatting:

Plot Stirling numbers of the second kind on a logarithmic scale:

Stirling numbers modulo 2:

Define a recursive function for generating set partitions:

Generate all set partitions of n elements:

Count the number of set partitions that have 1, 2, … n disjoint subsets:

The Stirling number of the second kind counts the number of disjoint subsets:

Closed form of derivatives of compositions with exponential functions:

A fair ‐sided die is thrown times independently. The probability that all faces appear at least once is given in terms of Stirling numbers of the second kind:

Plot the probability for a six-sided die:

Check with simulations:

Generate values from the ordinary generating function:

Generate values from the exponential generating function:

Stirling numbers of the second kind are effectively inverses of Stirling numbers of the first kind:

Calculate large Stirling numbers of the second kind using Cauchy's theorem:

Generate Stirling numbers of the second kind from the commutation relation :

The limit of finite differences of powers are Stirling numbers of the second kind:

Stirling numbers of the second kind are given by a partial Bell polynomial with unit arguments:

StirlingS2 can take large values for moderate‐size arguments:

The value at is defined to be 1: