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→Strict partition function: Use of small p and small q for function name (harmonized to previous sections) |
→Restricted partition function: odd or even part partition function, euler and glaisher theorems |
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More generally, if ''A'' is a set of natural numbers, we can consider partitions restricted to only elements in ''A''. The restricted partition function is then denoted ''p''<sub>''A''</sub>(''n''), or ''p''(''A'', ''n'').
Two important examples are the partitions restricted to only odd integer parts or only even integer parts, with the corresponding partition functions often denoted <math>p_o(n)</math> and <math>p_e(n)</math>.
Some general results on the asymptotic properties of the restricted partition function are known. If ''A'' possesses positive [[natural density]] α then▼
=== Euler and Glaisher's theorem ===
A theorem from Euler shows that the number of strict partitions is equal to the number of partitions with only odd parts: for all ''n'', <math>q(n) = p_o(n)</math>. This is generalized as [[Glaisher's theorem]], which states that the number of partitions with no more than ''d-1'' repetitions of any part is equal to the number of partitions with no part divisible by ''d''.
=== Asymptotics ===
▲Some general results on the asymptotic properties of
:<math> \log p_A(n) \sim C \sqrt{\alpha n} </math>
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