Partition function (number theory): Difference between revisions

Revision as of 09:15, 18 January 2023

Revision as of 09:37, 18 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''q''(''n'') gives the number of these strict partitions in relation to the given sum ''n''. Therefore the strict partition sequence Qq(n) satisfies the criterion Q''q(n) ≤ Pp(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.

=== ExemplaryExample values of strict partition numbers ===

Representations of the partitions:

{| class="wikitable"

|+ ExemplaryExample values of Qq(n) and associated number partitions

|-

! n || Qq(n) || Number partitionsPartitions without repeated addendsparts

!Number partitionsPartitions with only odd addendsparts

|-

| 0 || 1 || () empty partition

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|}

=== MacLaurin series ===

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

: <math>\sum_{k=0}^{\infty} Qq(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>

The following first addends are obtained:

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: <math>(x;x^2)_{\infty}^{-1} = 1+1x+1x^2+2x^3+2x^4+3x^5+4x^6+5x^7+6x^8+8x^9+10x^{10}...</math>

In comparison, the generating function of the regular partition numbers Pp(n) has this identity with respect to the theta function:

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

Important calculation formulas for the [[theta function]]:

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From this identity follows that formula:

: <math>\biggl[\sum_{n=0}^\infty Pp(n)x^n\biggr] = \biggl[\sum_{n=0}^\infty Pp(n)x^{2n}\biggr]\biggl[\sum_{n=0}^\infty Qq(n)x^n\biggr]</math>

Therefore those two formulas are valid for the synthesis of the number sequence Pp(n):

: <math>Pp(2n) = \sum_{k=0}^{n} Pp(n - k)Qq(2k)</math>

: <math>Pp(2n+1) = \sum_{k=0}^{n} Pp(n - k)Qq(2k + 1)</math>

In the following, two examples are accurately executed:

: <math>Pp(8) = \sum_{k=0}^{4} Pp(4 - k)Qq(2k) =</math>

: <math>= Pp(4)Qq(0) + Pp(3)Qq(2) + Pp(2)Qq(4) + Pp(1)Qq(6) + Pp(0)Qq(8) = </math>

: <math>= 5\times 1 + 3\times 1 + 2\times 2 + 1\times 4 + 1\times 6 = 22 </math>

: <math>Pp(9) = \sum_{k=0}^{4} Pp(4 - k)Qq(2k + 1) =</math>

: <math>= Pp(4)Q(1) + Pp(3)Q(3) + Pp(2)Qq(5) + Pp(1)Qq(7) + Pp(0)Qq(9) = </math>

: <math>= 5\times 1 + 3\times 2 + 2\times 3 + 1\times 5 + 1\times 8 = 30 </math>