Partition function (number theory): Difference between revisions

Revision as of 22:45, 3 January 2005

Revision as of 22:48, 3 January 2005

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The '''partition [[function (mathematics)|function]]''' p(''n'') represents the [[number]] of possible [[integer partition|partitions]] of a [[natural number]] ''n'', which is to say the number of distinct (and order independent) ways of representing ''n'' as a [[sum]] of natural numbers. By convention p(0) = 1, p(''n'') = 0 for ''n'' negative.

One way of getting a handle on the partition function involves an intermediate function p(''k'', ''n'') which represents the number of partitions of ''n'' using only natural numbers at least as large as ''k''. For any given value of ''k'', partitions counted by p(''k'',''n'') fit into exactly one of the following categories:

#smallest [[addend]] is ''k''

#smallest addend is strictly greater than ''k''

The number of partitions meeting the first condition is p(''k'', ''n''- − ''k''). To see this, imagine a list of all the partitions of the number ''n''- − ''k'' into numbers of size at least ''k'', then imagine appending "+''k''" to each partition in the list. Now what is it a list of?

The number of partitions meeting the second condition is p(''k'' + 1, ''n'') since a partition into parts of at least <i>''k</i>'' which contains no parts of at exactly <i>''k</i>'' must have all parts at least <i>''k'' + 1</i>.

Since the two conditions are [[mutually exclusive]], the number of partitions meeting either condition is p(''k'' + 1, ''n'') + p(''k'', ''n'' &minus; ''k''). The base cases of this [[recursion|recursive]] function are as follows:

* p(''k'', ''n'') = 0 if ''k'' > ''n''

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:p(6, 24) = '''16'''

Our original function p(<i>''n</i>'') is just p(1,<i> ''n</i>'').

An alternative computation of the partition function involves [[generating function|generating functions]]. Consider the infinite product

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The formulation of the generating function is similar to the product formulation of many [[modular form|modular forms]], giving some idea of the connection between the two. It can also be used in conjunction with the [[pentagonal number theorem]] to derive a recurrence for the partition function stating that

p(<i>k</i>) &minus; p(<i>''k</i>'' &minus; 1) &minus; p(''k'' &minus; 2) + p(''k'' &minus; 5) + p(''k'' &minus; 7) -&minus; p(''k'' &minus; 12) &minus; ... = 0,

where the sum is taken over all [[polygonal number|pentagonal numbers]] of the form ½''n''(3''n'' &minus; 1), including those where ''n'' < 0, and the terms continue to alternate +, +, &minus;, &minus;, +, +, ...