### Abstract

The theory's main theorem states that the cardinality of set partitions of size k on a carrier set of size n is expressed by Stirling numbers of the second kind. In Isabelle, Stirling numbers of the second kind are defined in the AFP entry `Discrete Summation` through their well-known recurrence relation. The main theorem relates them to the alternative definition as cardinality of set partitions. The proof follows the simple and short explanation in Richard P. Stanley's `Enumerative Combinatorics: Volume 1` and Wikipedia, and unravels the full details and implicit reasoning steps of these explanations.

### License

BSD License

### Topics

- Mathematics/Combinatorics

### Session Card_Partitions

- Set_Partition
- Injectivity_Solver
- Card_Partitions

### Depends on

- Discrete Summation

### Used by

- Hypergraph Colouring Bounds
- General Probabilistic Techniques for Combinatorics and the Lovasz Local Lemma
- Combinatorial Design Theory
- The Falling Factorial of a Sum
- The Twelvefold Way
- Spivey’s Generalized Recurrence for Bell Numbers

## Cite

×

Card_Partitions-AFP

    @article{Card_Partitions-AFP,
      author  = {Lukas Bulwahn},
      title   = {Cardinality of Set Partitions},
      journal = {Archive of Formal Proofs},
      month   = {December},
      year    = {2015},
      note    = {\url{https://isa-afp.org/entries/Card_Partitions.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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Older releases:

- Sep 13, 2023 : Isabelle2023
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- Dec 14, 2021 : Isabelle2021-1
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