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OpenMath Content Dictionary: combinat1

Canonical URL:
http://www.openmath.org/cd/combinat1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
combinat1.ocd
CD as XML Encoded OpenMath:
combinat1.omcd
Defines:
Bell, Fibonacci, Stirling1, Stirling2, binomial, multinomial
Date:
2004-03-30
Version:
3
Review Date:
2006-03-30
Status:
experimental


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This CD defines some basic combinatorics definitions.

Written by S. Dalmas (INRIA Sophia Antipolis) for the Esprit OpenMath
project. 

binomial

Role:
application
Description:

The binomial coefficients. binomial(n, m) is the number of ways of choosing m objects from a collection of n distinct objects without regard to the order.

Commented Mathematical property (CMP):
binomial(n,m) = n!/(m!*(n-m)!)
Formal Mathematical property (FMP):
n m = n ! m ! ( n - m ) !
Example:
4 2 = 6
Signatures:
sts


[Next: multinomial] [Last: Bell] [Top]

multinomial

Role:
application
Description:

The multinomial coefficient, multinomial(n, n1, ... nk) is the number of ways of choosing ni objects of type i (i from 1 to k) without regard to order, in such a way that the total number of objects chosen is n. multinomial(n, n1, ... nk) is equal to n!/(n1!*n2! ...*nk!).

Commented Mathematical property (CMP):
multinomial(n, n1, ... nk) is equal to n!/(n1!*n2! ...*nk!) where n=n1+...+nk
Formal Mathematical property (FMP):
apply_to_list ( cons ( n , nList ) , cons ( n , nList ) ) = n ! apply_to_list ( × , nList 2 ) apply_to_list ( vector , nlist 2 ) i = apply_to_list ( vector , nList ) i ! n = apply_to_list ( + , nList )
Example:
8 2 3 3 = 560
Signatures:
sts


[Next: Stirling1] [Previous: binomial] [Top]

Stirling1

Role:
application
Description:

The Stirling numbers of the first kind. (-1)^(n-m)*Stirling1(n,m) is the number of permutations of n symbols which have exactly m cycles. Note that there are a few slightly different definitions of these numbers.

Commented Mathematical property (CMP):
Stirling1(n,m) = the sum k=0 to n-m of (-1)^k * binomial(n-1+k, n-m+k) * binomial(2n-m,n-m-k) * Stirling2(n,m)
Formal Mathematical property (FMP):
Stirling1 ( n , m ) = k = 0 n - m - 1 k n - 1 + k n - m + k 2 n - m n - m - k Stirling2 ( n , m )
Example:
Stirling1 ( 10 , 7 ) = -9450
Signatures:
sts


[Next: Stirling2] [Previous: multinomial] [Top]

Stirling2

Role:
application
Description:

The Stirling numbers of the second kind. Stirling2(n, m) is the number of partitions of a set with n elements into m non empty subsets. Note that there are a few slightly different definitions of these numbers.

Commented Mathematical property (CMP):
Stirling2(n,m) = 1/m! * the sum from k=0 to m of (-1)^(m-k) * binomial(m,k) * k^n
Formal Mathematical property (FMP):
Stirling2 ( n , m ) = 1 m ! k = 0 m - 1 ( m - k ) m k k n
Example:
Stirling2 ( 7 , 3 ) = 301
Signatures:
sts


[Next: Fibonacci] [Previous: Stirling1] [Top]

Fibonacci

Role:
application
Description:

The Fibonacci numbers, defined by the linear recurrence: Fibonacci(0) = 0, Fibonacci(1) = 1, and Fibonacci(n + 1) = Fibonacci(n) + Fibonacci(n - 1). Note that some authors define Fibonacci(0) = 1.

Commented Mathematical property (CMP):
Fibonacci(0) = 0, Fibonacci(1) = 1, and Fibonacci(n + 1) = Fibonacci(n) + Fibonacci(n - 1)
Formal Mathematical property (FMP):
Fibonacci ( 0 ) = 0 Fibonacci ( 1 ) = 1 Fibonacci ( n + 1 ) = Fibonacci ( n ) + Fibonacci ( n - 1 )
Example:
Fibonacci ( 10 ) = 55
Signatures:
sts


[Next: Bell] [Previous: Stirling2] [Top]

Bell

Role:
application
Description:

The Bell numbers: Bell(n) is the total number of possible partitions of a set of n elements.

Commented Mathematical property (CMP):
Bell(n) = the sum from k=0 to n of Stirling2(n,k)
Formal Mathematical property (FMP):
Bell ( n ) = k = 0 n Stirling2 ( n , k )
Example:
Bell ( 7 ) = 877
Signatures:
sts


[First: binomial] [Previous: Fibonacci] [Top]

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