Partition function (number theory): Difference between revisions

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=== Definition and properties ===

A partition in which no part occurs more than one is called ''strict'', or is said to be a partition ''into distinct parts''. The function ''q''(''n'') gives the number of these strict partitions of the given sum ''n''. For example, ''q''(3) = 2 because the partitions 3 and 2 + 1 are strict, while the third partition 1 + 1 + 1 of 3 has repeated parts. The number ''q''(''n'') is also equal to the number of partitions of ''n'' in which only odd summands are permitted.<ref>{{cite book|first=Richard P.|last=Stanley|author-link=Richard P. Stanley|title=Enumerative Combinatorics 1 |series=Cambridge Studies in Advanced Mathematics|volume=49|publisher=Cambridge University Press|isbn=0-521-66351-2 |year=1997|loc=Proposition 1.8.5}}</ref>

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 ===