From: Bill Marshall Subject: Re: Ways to express integer as sum of integers Date: Sat, 23 Dec 2000 08:29:34 GMT Newsgroups: sci.math Summary: [missing] In article , superfaris@hotmail.com (Funky Llama) wrote: > Hi, > > In how many ways can you express an integer as a sum of other > integers, where the order doesn't matter, e.g. 4 -> > {4,3+1,2+2,2+1+1,1+1+1+1} so W(4) = 5... what is the general > expression for W(n) and how do you come by it? > > Ta, > FL > These are known as partition numbers. Let W(n,p) be the partition of n into p parts. Then W(n,p) = W(n-1,p-1)+W(n-p,p). An approximate formula for W(n) was guessed by the Indian mathematician Srinivasa Ramanujan and proven by Ramanujan and Hardy: e^(PI sqrt(2n/3)) W(n) ~= ----------------- 4n sqrt(3) An exact formula was later derived by Rademacher but it is horribly complicated. For references go to Sloane's online Encyclopedia of Number Sequences at http://www.research.att.com/~njas/sequences/index.html and enter the following sequence in the search field: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101 Bill Marshall Sent via Deja.com http://www.deja.com/ ============================================================================== From: Qqqquet@mindspring.com (Leroy Quet) Subject: RE:Ways to express integer as sum of integers Date: 23 Dec 2000 23:41:37 -0500 Newsgroups: sci.math Funky Llama wrote: >Hi, >In how many ways can you express an integer as a sum of other >integers, where the order doesn't matter, e.g. 4 -> >{4,3+1,2+2,2+1+1,1+1+1+1} so W(4) = 5... what is the general >expression for W(n) and how do you come by it? W(n) is called the number of (unrestricted) partitions of n. It is usually denoted as p(n). p(n) satisfies the generating function: 1 + sum_{n=1}^infinity[p(n) *x^m] = product_{n=1}^infinity [1/(1 -x^n)]. There has been a lot written about partitions in the past. I can't think now of any specific books on the subject, though I know lots of books have been written on partitions. Thanks, Leroy Quet (Spam Block: Extra Q added.)