
Spivey's
Generalized
Recurrence
for
Bell
Numbers
Title: 
Spivey's Generalized Recurrence for Bell Numbers 
Author:

Lukas Bulwahn (lukas /dot/ bulwahn /at/ gmail /dot/ com)

Submission date: 
20160504 
Abstract: 
This entry defines the Bell numbers as the cardinality of set partitions for
a carrier set of given size, and derives Spivey's generalized recurrence
relation for Bell numbers following his elegant and intuitive combinatorial
proof.
As the set construction for the combinatorial proof requires construction of
three intermediate structures, the main difficulty of the formalization is
handling the overall combinatorial argument in a structured way.
The introduced proof structure allows us to compose the combinatorial argument
from its subparts, and supports to keep track how the detailed proof steps are
related to the overall argument. To obtain this structure, this entry uses set
monad notation for the set construction's definition, introduces suitable
predicates and rules, and follows a repeating structure in its Isar proof. 
BibTeX: 
@article{Bell_Numbers_SpiveyAFP,
author = {Lukas Bulwahn},
title = {Spivey's Generalized Recurrence for Bell Numbers},
journal = {Archive of Formal Proofs},
month = may,
year = 2016,
note = {\url{http://isaafp.org/entries/Bell_Numbers_Spivey.shtml},
Formal proof development},
ISSN = {2150914x},
}

License: 
BSD License 
Depends on: 
Card_Number_Partitions, Card_Partitions 
Used by: 
Card_Equiv_Relations, Twelvefold_Way 
Status: [ok] 
This is a development version of this entry. It might change over time
and is not stable. Please refer to release versions for citations.

