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Partition function

Table of contents: Specific values - Generating functions - Sums and recurrence relations - Congruences - Inequalities - Asymptotic expansions - Hardy-Ramanujan-Rademacher formula

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Symbol: PartitionsP p(n)p(n) Integer partition function
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p(n)=A000041 ⁣(n)p(n) = \text{A000041}\!\left(n\right)

Specific values

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Table of p(n)p(n) for 0n2000 \le n \le 200
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p(0)=#{[]}=1p(0) = \# \left\{\left[\right]\right\} = 1
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p(1)=#{[1]}=1p(1) = \# \left\{\left[1\right]\right\} = 1
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p(2)=#{[2],[1,1]}=2p(2) = \# \left\{\left[2\right], \left[1, 1\right]\right\} = 2
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p(3)=#{[3],[2,1],[1,1,1]}=3p(3) = \# \left\{\left[3\right], \left[2, 1\right], \left[1, 1, 1\right]\right\} = 3
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p(4)=#{[4],[3,1],[2,2],[2,1,1],[1,1,1,1]}=5p(4) = \# \left\{\left[4\right], \left[3, 1\right], \left[2, 2\right], \left[2, 1, 1\right], \left[1, 1, 1, 1\right]\right\} = 5
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p ⁣(n)=0p\!\left(-n\right) = 0
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Table of p ⁣(10n)p\!\left({10}^{n}\right) to 50 digits for 0n300 \le n \le 30

Generating functions

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n=0p(n)qn=1ϕ(q)\sum_{n=0}^{\infty} p(n) {q}^{n} = \frac{1}{\phi(q)}

Sums and recurrence relations

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p(n)=k=1n+1(1)k+1(p ⁣(nk(3k1)2)+p ⁣(nk(3k+1)2))p(n) = \sum_{k=1}^{n + 1} {\left(-1\right)}^{k + 1} \left(p\!\left(n - \frac{k \left(3 k - 1\right)}{2}\right) + p\!\left(n - \frac{k \left(3 k + 1\right)}{2}\right)\right)
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p(n)=1nk=0n1σ1 ⁣(nk)p(k)p(n) = \frac{1}{n} \sum_{k=0}^{n - 1} \sigma_{1}\!\left(n - k\right) p(k)

Congruences

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p ⁣(5n+4)0(mod5)p\!\left(5 n + 4\right) \equiv 0 \pmod {5}
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p ⁣(7n+5)0(mod7)p\!\left(7 n + 5\right) \equiv 0 \pmod {7}
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p ⁣(11n+6)0(mod11)p\!\left(11 n + 6\right) \equiv 0 \pmod {11}

Inequalities

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p(n)p ⁣(n+1)p(n) \le p\!\left(n + 1\right)
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p(n)<p ⁣(n+1)p(n) < p\!\left(n + 1\right)
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p(n)np(n) \ge n
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p(n)2np(n) \le {2}^{n}

Asymptotic expansions

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p(n)eπ2n/34n3,  np(n) \sim \frac{{e}^{\pi \sqrt{2 n / 3}}}{4 n \sqrt{3}}, \; n \to \infty

Hardy-Ramanujan-Rademacher formula

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p(n)=2π(24n1)3/4k=1A ⁣(n,k)kI3/2 ⁣(πk23(n124))p(n) = \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{\infty} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)
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Symbol: HardyRamanujanA A ⁣(n,k)A\!\left(n, k\right) Exponential sum in the Hardy-Ramanujan-Rademacher formula
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A ⁣(n,k)=r=0k1δ(gcd(r,k),1)exp ⁣(πi(s ⁣(r,k)2nrk))A\!\left(n, k\right) = \sum_{r=0}^{k - 1} \delta_{(\gcd\left(r, k\right),1)} \exp\!\left(\pi i \left(s\!\left(r, k\right) - \frac{2 n r}{k}\right)\right)
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p(n)2π(24n1)3/4k=1NA ⁣(n,k)kI3/2 ⁣(πk23(n124))44π22253N+π275Nn1sinh ⁣(πN2n3)\left|p(n) - \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{N} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)\right| \le \frac{44 {\pi}^{2}}{225 \sqrt{3 N}} + \frac{\pi \sqrt{2}}{75} \sqrt{\frac{N}{n - 1}} \sinh\!\left(\frac{\pi}{N} \sqrt{\frac{2 n}{3}}\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-11 15:50:15.016492 UTC