Partition function (number theory): Difference between revisions

Revision as of 08:55, 13 June 2022

Revision as of 08:40, 24 June 2022

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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 ℕ₀\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.

=== Exemplary values of strict partition numbers ===

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=== MacLaurin Series ===

The corresponding generating function based on the [[MacLaurin series]] with the numbers Q(n) as coefficients in front of xⁿxⁿ is as follows:

: <math>\sum_{k=0}^{\infty} Q(k)x^k = (x;x^2)_{\infty}^{-1} = \vartheta_{00}(x)^{1/6}\vartheta_{01}(x)^{-1/3}\biggl\{\frac{1}{16\,x}\bigl[\vartheta_{00}(x)^4 - \vartheta_{01}(x)^4\bigr]\biggr\}^{1/24}</math>