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Stirling numbers

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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
How to Cite This Entry:
Stirling numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stirling_numbers&oldid=54253