Partition Function Calculator
Free online partition function calculator. Enter your values and get instant results with the formula, step-by-step working, a worked example and FAQs — everything runs locally in your browser on Calcton, no signup required.
中文版：整数划分数在线计算器
划分数 p(n)：把 n 写成正整数无序和的方案数。p(4) = 5：4、3+1、2+2、2+1+1、1+1+1+1。
欧拉五边形数定理的递推
p(n) = Σ (−1)^(k−1)·[p(n − k(3k−1)/2) + p(n − k(3k+1)/2)]，广义五边形数 1, 2, 5, 7, 12, 15, …
生成函数 1/((1−x)(1−x²)(1−x³)…) 系数即 p(n)；欧拉发现 (1−x)(1−x²)(1−x⁵)(1−x⁷)… 与它互为倒数——五边形数定理——从而得到 O(n√n) 递推。Hardy 与 Ramanujan 给出渐近公式，Rademacher 进一步改成精确级数。拉马努金同余 p(5k+4) ≡ 0 (mod 5)、p(7k+5) ≡ 0 (mod 7)、p(11k+6) ≡ 0 (mod 11) 至今仍在被研究推广。应用：整数拆分计数、杨图表示论、统计力学配分、密码学随机性检验。
How to use the Partition Function Calculator
- Fill in the inputs with your own numbers — every field shows what it expects.
- The result updates instantly as you type; no "Calculate" click needed.
- Read the result card for the main figure, the step-by-step formula, and practical notes.
Frequently asked questions
Is the Partition Function Calculator free?
Yes — all math calculators on Calcton are free, with no sign-up required. Your inputs are processed in your browser and never sent to a server.
Is there a Chinese version of this tool?
Yes. Use the Chinese version link above — it is the same calculator with the interface in Simplified Chinese.
References
Formulas and benchmarks used by this calculator follow these published sources.
KevinContent LeadCalcton Editorial Team · Last updated 2026-09-26
Kevin maintains Calcton's formula verification and content: every calculator cites the standard it implements, worked examples are recomputed programmatically, and health benchmarks follow WHO and international guidelines.
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