Partitioning the Integers
Date: 15 Mar 1995 19:46:08 -0500 From: Anonymous Subject: partitions I'm a high school math teacher, and I recently assigned my discrete math students a project. One of my students chose the topic of partitions of the positive integers. We have studied various counting models, one of which is combinations with repetitions. Are you aware of a book which treats the subject of partitions, or do you know of an elementary approach to the solution which I could explore with my students? They know how to find, for example, the number of four digit numbers, the sum of whose digits is 9, and I am convinced that the problem of partitioning the integers is related. Thanks a lot!
Date: 16 Mar 1995 16:00:36 -0500 From: Stephen B Maurer) (by way of email@example.com Stephen Weimar Subject: Re: partitions First, it depends on exactly what you mean by partitions. By the partitions of 4, mathematicians usually mean 4 3 1 2 2 2 1 1 1 1 1 1 If order is considered, then the partitions are called ordered partitions. Here they are for 4: 4 3 1 1 3 2 2 2 1 1 1 2 1 1 1 2 1 1 1 1 The latter objects are indeed closely related to combinations with repetitions, but (unordered) partitions are a different story. There are no general, exact counting formulas for them, though there are many asymptotic (limit) formulas, and many many theorems of the form "the number of partitions of this form = the number of partitions of that form". Any book on number theory will have at least one chapter on partitions. Combinatorics books typically spend less time on them. Hope this helps. Keep up the good work. Stephen B Maurer Professor of Mathematics Swarthmore College
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