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Let p¯k(n) be the number of partitions of n with largest part at most k which is equivalent to partition into at most k parts. I do know an expression for that function. ( product of 1/1(1-n) through 1/(1-n)^k )

What I am searching for is an expression that gives that number of partitions under both restrictions, so all partitions of n that have at most k parts AND all parts <= j.

blues
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  • Although at first glance these may seem slightly different problems, in my answer I show the equivalent transformations for counting (1) partitions with exactly $k$ distinct parts, each at most $M$, (2) partitions with exactly $k$ parts (not necessarily distinct), each at most $M$, and (3) partitions of at most $k$ parts, each at most $M$. – hardmath Jan 03 '16 at 02:57

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