Partition function (number theory): Difference between revisions

Revision as of 09:19, 19 January 2023

Revision as of 15:25, 19 January 2023

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=== Definition and properties ===

If no summand occurs repeatedly<ref>{{cite web|title=code golf -– Strict partitions of a positive integer|periodical=|publisher=|url=https://codegolf.stackexchange.com/questions/71941/strict-partitions-of-a-positive-integer|url-status=|format=|access-date=2022-03-09|archive-url=|archive-date=|last=|date=|year=|language=|pages=|quote=}}</ref> in the affected partition sums, then the so called strict partitions are present. The function ''q''(''n'') gives the number of these strict partitions in relation to the given sum ''n''. Therefore the strict partition sequence q(n) satisfies the criterion ''q(n) ≤ p(n)'' for all <math>n \isin \mathbb{N}_0</math>. The same result<ref>{{cite web|title=A000009 -– OEIS|periodical=|publisher=|url=https://oeis.org/A000009|url-status=|format=|access-date=2022-03-09|archive-url=|archive-date=|last=|date=|year=|language=|pages=|quote=}}</ref> results if only odd summands<ref>{{cite web|title=Partition Function Q|periodical=|publisher=|url=https://mathworld.wolfram.com/|url-status=|format=|access-date=2022-03-09|archive-url=|archive-date=|last=Eric W. Weisstein|date=|year=|language=en|pages=|quote=}}</ref> may appear in the partition sum, but these may also occur more than once.

=== Example values of strict partition numbers ===

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If ''A'' possesses positive [[natural density]] α then <math> \log p_A(n) \sim C \sqrt{\alpha n}</math>, with <math>C = \pi\sqrt\frac23</math>

and conversely if this asymptotic property holds for ''p''_''A''(''n'') then ''A'' has natural density α.{{sfn|Nathanson|2000|pp=475-85475–85}} This result was stated, with a sketch of proof, by Erdős in 1942.<ref name=erdos42></ref>{{sfn|Nathanson|2000|p=495}}

If ''A'' is a finite set, this analysis does not apply (the density of a finite set is zero). If ''A'' has ''k'' elements whose greatest common divisor is 1, then{{sfn|Nathanson|2000|pp=458-64458–64}}

:<math> p_A(n) = \left(\prod_{a \in A} a^{-1}\right) \cdot \frac{n^{k-1}}{(k-1)!} + O(n^{k-2}) . </math>