36. Partitions | |||
---|---|---|---|

Home | Menu | Previous | Next |

Partitions (Basics)

Divide an integer into any number of positive integral parts. This is aNUMBER | PARTITIONS | NUMBER OF PARTITIONS |

n | p(n) | |

1 | 1 | p(1) = 1 |

2 | 1+1, 2 | p(2) = 2 |

3 | 1+1+1, 1+2, 3 | p(3) = 3 |

4 | 1+1+1+1, 1+1+2, 1+3, 2+2, 4 | p(4) = 5 |

5 | 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+4, 2+2+1, 2+3, 5 | p(5) = 7 |

6 | p(6) = 11 | |

7 | p(7) = 15 | |

... | ... | |

20 | p(20) = 627 | |

... | ... | |

50 | p(50) = 204,226 |

1. | As n increases, .p(n) increases rapidly | |

2. | Look at the partitions of 5:p(5): 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+4, 2+2+1, 2+3, 5Of these 1+1+1+1+1, 1+1+3, 2+2+1 are ODD PARTS as there are odd number of parts And 1+4, 2+3, 5 are DISTINCT PARTS as no part is repeated. has 3 distinct parts and also, p(5) has 3 odd parts.p(5) | |

3. | states: In partitions of any number, Euler's TheoremDistinct parts equal Odd parts. | |

4. | Is it possible to say if p(n) is odd or even, for any n? | |

5. | In the Theory of Partitions, Hardy and Ramanujan contributed a remakable theorem. |