Stirling numbers
2020 Mathematics Subject Classification: Primary: 05A Secondary: 11B73 [MSN][ZBL]
In combinatorics, counts of certain arrangements of objects into a given number of structures. There are two kinds of Stirling number, depending on the nature of the structure being counted.
- The Stirling numbers of the first kind count the number of ways n labelled objects can be arranged into k cycles: cycles are regarded as equivalent, and counted only once, if they differ by a cyclic permutation, thus [ABC] = [BCA] = [CAB] but is counted as different from [CBA] = [BAC] = [ACB]. The order of the cycles in the list is irrelevant.
For example, 4 objects can be arranged into 2 cycles in eleven ways, so S(4,2) = 11.
- [ABC],[D]
- [ACB],[D]
- [ABD],[C]
- [ADB],[C]
- [ACD],[B]
- [ADC],[B]
- [BCD],[A]
- [BDC],[A]
- [AB],[CD]
- [AC],[BD]
- [AD],[BC]
The number of permutations on n letters with exactly k cycles is given by S(n,k). Hence n!=\sum_{k=1}^nS(n,k) \ .
- The Stirling numbers of the second kind s(n,k) count the number of ways n labelled objects can be arranged into k subsets: the subsets are regarded as equivalent, and counted only once, if they have the same elements, thus \{ABC\} = \{BCA\} = \{CAB\} = \{CBA\} = \{BAC\} = \{ACB\}; the order of the subsets in the list is also irrelevant.
For example, 4 objects can be arranged into 2 subsets in seven ways, so s(4,2) = 7:
- {ABC},{D}
- {ABD},{C}
- {ACD},{B}
- {BCD},{A}
- {AB},{CD}
- {AC},{BD}
- {AD},{BC}
If B_n denotes the Bell numbers, the total number of partitions of a set of size n, thren B_n = \sum_{k=1}^n s(n,k) \ .
See also: Combinatorial analysis.
References
- Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, Addison Wesley (1989) pp.243-253. ISBN 0-201-14236-8
Stirling numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stirling_numbers&oldid=54253