Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 4 / a / ii

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- Solution ii

Solution

Subtract one from each of the

e

positive parts. This bijects partitions into exactly

e

parts with partitions into at most

e

parts and decreases the size by

e

. The required generating function is therefore

\frac{x ^{e}}{( 1 - x ) ( 1 - x ^{2} ) \dots ( 1 - x ^{e} )} .

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