Editing Integer partition (section)

===Restricted part size or number of parts===
{{main|Triangle of partition numbers}}
By taking conjugates, the number {{math|''p''<sub>''k''</sub>(''n'')}} of partitions of {{math|''n''}} into exactly ''k'' parts is equal to the number of partitions of {{math|''n''}} in which the largest part has size {{math|''k''}}.  The function {{math|''p''<sub>''k''</sub>(''n'')}} satisfies the recurrence
: {{math|1=''p''<sub>''k''</sub>(''n'') = ''p''<sub>''k''</sub>(''n'' − ''k'') + ''p''<sub>''k''−1</sub>(''n'' &minus; 1)}}
with initial values {{math|1=''p''<sub>0</sub>(0) = 1}} and {{math|1=''p''<sub>''k''</sub>(''n'') = 0}} if {{math|''n'' &le; 0 or ''k'' &le; 0}} and {{math|''n''}} and {{math|''k''}} are not both zero.<ref>Richard Stanley, ''Enumerative Combinatorics'', volume 1, second edition.  Cambridge University Press, 2012.  Chapter 1, section 1.7.</ref>

One recovers the function ''p''(''n'') by

:<math>
p(n) = \sum_{k = 0}^n p_k(n).
</math>

One possible generating function for such partitions, taking ''k'' fixed and ''n'' variable, is 
: <math> \sum_{n \geq 0} p_k(n) x^n = x^k\prod_{i = 1}^k \frac{1}{1 - x^i}.</math>

More generally, if ''T'' is a set of positive integers then the number of partitions of ''n'', all of whose parts belong to ''T'', has [[generating function]] 
:<math>\prod_{t \in T}(1-x^t)^{-1}.</math>
This can be used to solve [[change-making problem]]s (where the set ''T'' specifies the available coins).  As two particular cases, one has that the number of partitions of ''n'' in which all parts are 1 or 2 (or, equivalently, the number of partitions of ''n'' into 1 or 2 parts) is
:<math>\left \lfloor \frac{n}{2}+1 \right \rfloor ,</math>
and the number of partitions of ''n'' in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of ''n'' into at most three parts) is the nearest integer to (''n'' + 3)<sup>2</sup> / 12.<ref>{{cite book|last=Hardy|first=G.H.|title=Some Famous Problems of the Theory of Numbers|url=https://archive.org/details/in.ernet.dli.2015.84630|publisher=Clarendon Press|date=1920}}</ref>