This document is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. The copyright holder grants you permission to redistribute this document freely as a verbatim copy. Furthermore, the copyright holder permits you to develop any derived work from this document provided that the following conditions are met. a) The derived work acknowledges the fact that it is derived from this document, and maintains a prominent reference in the work to the original source. b) The fact that the derived work is not the original OpenMath document is stated prominently in the derived work. Moreover if both this document and the derived work are Content Dictionaries then the derived work must include a different CDName element, chosen so that it cannot be confused with any works adopted by the OpenMath Society. In particular, if there is a Content Dictionary Group whose name is, for example, `math' containing Content Dictionaries named `math1', `math2' etc., then you should not name a derived Content Dictionary `mathN' where N is an integer. However you are free to name it `private_mathN' or some such. This is because the names `mathN' may be used by the OpenMath Society for future extensions. c) The derived work is distributed under terms that allow the compilation of derived works, but keep paragraphs a) and b) intact. The simplest way to do this is to distribute the derived work under the OpenMath license, but this is not a requirement. If you have questions about this license please contact the OpenMath society at http://www.openmath.org. combinat1 http://www.openmath.org/cd http://www.openmath.org/cd/combinat1.ocd 2006-03-30 2004-03-30 3 0 experimental This CD defines some basic combinatorics definitions. Written by S. Dalmas (INRIA Sophia Antipolis) for the Esprit OpenMath project. binomial application 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. binomial(n,m) = n!/(m!*(n-m)!) 4 2 6 multinomial application 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!). multinomial(n, n1, ... nk) is equal to n!/(n1!*n2! ...*nk!) where n=n1+...+nk 8 2 3 3 560 Stirling1 application 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. 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) 2 10 7 -9450 Stirling2 application 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. Stirling2(n,m) = 1/m! * the sum from k=0 to m of (-1)^(m-k) * binomial(m,k) * k^n 7 3 301 Fibonacci application 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. Fibonacci(0) = 0, Fibonacci(1) = 1, and Fibonacci(n + 1) = Fibonacci(n) + Fibonacci(n - 1) 10 55 Bell application The Bell numbers: Bell(n) is the total number of possible partitions of a set of n elements. Bell(n) = the sum from k=0 to n of Stirling2(n,k) 7 877