Editing Integer partition (section)

===Partitions in a rectangle and Gaussian binomial coefficients===
{{Main|Gaussian binomial coefficient}}
One may also simultaneously limit the number and size of the parts.  Let {{math|''p''(''N'', ''M''; ''n'')}} denote the number of partitions of {{mvar|n}} with at most {{mvar|M}} parts, each of size at most {{mvar|N}}.  Equivalently, these are the partitions whose Young diagram fits inside an {{math|''M'' × ''N''}} rectangle.  There is a recurrence relation
<math display=block>p(N,M;n) = p(N,M-1;n) + p(N-1,M;n-M)</math>
obtained by observing that <math>p(N,M;n) - p(N,M-1;n)</math> counts the partitions of {{mvar|n}} into exactly {{mvar|M}} parts of size at most {{mvar|N}}, and subtracting 1 from each part of such a partition yields a partition of {{math|''n'' − ''M''}} into at most {{mvar|M}} parts.{{sfn|Andrews|1976|pp=33–34}}

The Gaussian binomial coefficient is defined as:
<math display=block>{k+\ell \choose \ell}_q = {k+\ell \choose k}_q = \frac{\prod^{k+\ell}_{j=1}(1-q^j)}{\prod^{k}_{j=1}(1-q^j)\prod^{\ell}_{j=1}(1-q^j)}.</math>
The Gaussian binomial coefficient is related to the [[generating function]] of {{math|''p''(''N'', ''M''; ''n'')}} by the equality
<math display=block>\sum^{MN}_{n=0}p(N,M;n)q^n = {M+N \choose M}_q.</math>