I'm looking for a closed-form formula for the number of partitions of integer $n$ into integer parts less than or equal to 9. Thanks.
2026-03-29 19:17:34.1774811854
Number of partitions into parts not greater than 9
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One can show that the number $p_k(n)$ of partitions of $n$ into exactly $k$ parts is equal to the number of partitions of $n$ in which the largest part has size $k$. So you are looking for a formula for $p_9(k)$. Rubinstein has given an explicit formula for $p_k(n)$ in terms of Bernoulli polynomials, see here. A. Sills has given Rademacher-type formulas for the restricted partition function.