Partition function (number theory): Difference between revisions

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{{Short description|The number of partitions of an integer}}

[[File:Ferrer partitioning diagrams.svg|thumb|The values <math>p(1), \dots, p(8)</math> of the partition function (1, 2, 3, 5, 7, 11, 15, and 22) can be determined by counting the [[Young diagram]]s for the partitions of the numbers from 1 to 8.]]

In [[number theory]], the '''partition function''' {{math|''p''(''n'')}} represents the [[number]] of possible [[PartitionInteger (number theory)partition|partitions]] of a non-negative integer {{mvar|n}}. For instance, {{math|1=''p''(4) = 5}} because the integer 4 has the five partitions {{math|1 + 1 + 1 + 1}}, {{math|1 + 1 + 2}}, {{math|1 + 3}}, {{math|2 + 2}}, and {{math|4}}.

No [[closed-form expression]] for the partition function is known, but it has both [[asymptotic analysis|asymptotic expansions]] that accurately approximate it and [[recurrence relation]]s by which it can be calculated exactly. It grows as an [[exponential function]] of the [[square root]] of its argument. The [[multiplicative inverse]] of its [[generating function]] is the [[Euler function]]; by Euler's [[pentagonal number theorem]] this function is an alternating sum of [[pentagonal number]] powers of its argument.