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Let $\bar{n} = (n_1,\dots, n_k)$ be a fixed tuple of positive integers. Define $\mathcal P_m ({\bar{n}}) $ to be the set of all partitions of $m$ with exactly $k$ parts and the i-th part is bounded by $n_i$.

How to find the cardinality of the set $\mathcal P_m ({\bar{n}}) $?

Kindly share your thoughts.

Sebastiano
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GA316
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  • Two big questions that needs to be answered: (1) Is $\overline{n}$ requires to be made of distinct $n_i$? (2) Does order matter? For example, take $\overline{n}= {3,5,2,5}$. Is ${1,2,1,1}$ the same as ${1,1,1,2}$? And are those two the same as ${2,1,1,1}$? ---- Also I want to say that just the problem of partitioning into exactly $k$ parts is mildly difficult; this count seems likely to be extremely difficult. – Eric Snyder Jan 29 '23 at 09:27
  • @EricSnyder (1) $\bar{n}$ need not be distinct in my case. (2) Yes. Order Matters. (Does it means that should I consider the compositions?) In my case, the partitions ${1,2,1,1}$ and ${1,1,1,2}$ are distinct. Thank you. – GA316 Jan 29 '23 at 10:11

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