gives the Stirling number of the second kind .
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- gives the number of ways of partitioning a set of elements into non‐empty subsets.
- StirlingS2 automatically threads over lists.
Examplesopen allclose all
Plot Stirling numbers of the second kind on a logarithmic scale:
Stirling numbers modulo 2:
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:
Properties & Relations (6)
Generate values from the 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:
Possible Issues (2)
StirlingS2 can take large values for moderate‐size arguments:
The value at is defined to be 1: