stirling2 — SciPy v1.15.2 Manual (original) (raw)
scipy.special.
scipy.special.stirling2(N, K, *, exact=False)[source]#
Generate Stirling number(s) of the second kind.
Stirling numbers of the second kind count the number of ways to partition a set with N elements into K non-empty subsets.
The values this function returns are calculated using a dynamic program which avoids redundant computation across the subproblems in the solution. For array-like input, this implementation also avoids redundant computation across the different Stirling number calculations.
The numbers are sometimes denoted
\[{N \brace{K}}\]
see [1] for details. This is often expressed-verbally-as “N subset K”.
Parameters:
Nint, ndarray
Number of things.
Kint, ndarray
Number of non-empty subsets taken.
exactbool, optional
Uses dynamic programming (DP) with floating point numbers for smaller arrays and uses a second order approximation due to Temme for larger entries of N and K that allows trading speed for accuracy. See [2] for a description. Temme approximation is used for values n>50. The max error from the DP has max relative error4.5*10^-16 for n<=50 and the max error from the Temme approximation has max relative error 5*10^-5 for 51 <= n < 70 and9*10^-6 for 70 <= n < 101. Note that these max relative errors will decrease further as n increases.
Returns:
valint, float, ndarray
The number of partitions.
See also
The number of combinations of N things taken k at a time.
Notes
- If N < 0, or K < 0, then 0 is returned.
- If K > N, then 0 is returned.
The output type will always be int or ndarray of object. The input must contain either numpy or python integers otherwise a TypeError is raised.
References
[1]
R. L. Graham, D. E. Knuth and O. Patashnik, “Concrete Mathematics: A Foundation for Computer Science,” Addison-Wesley Publishing Company, Boston, 1989. Chapter 6, page 258.
[2]
Temme, Nico M. “Asymptotic estimates of Stirling numbers.” Studies in Applied Mathematics 89.3 (1993): 233-243.
Examples
import numpy as np from scipy.special import stirling2 k = np.array([3, -1, 3]) n = np.array([10, 10, 9]) stirling2(n, k) array([9330.0, 0.0, 3025.0])