## Abstract

For two sets A and M of positive integers and for a positive integer n, let p(n, A, M) denote the number of partitions of n with parts in A and multiplicities in M, that is, the number of representations of n in the form n = ∑a∈Am_{a}a where m_{a} ∈ M∪{0} for all a, and all numbers ma but finitely many are 0. It is shown that there are infinite sets A and M so that p(n, A, M) = 1 for every positive integer n. This settles (in a strong form) a problem of Canfield and Wilf. It is also shown that there is an infinite set M and constants c and n0 so that for A = {k!}_{k≥1} or for A = {k^{k}}k≥1, 0 < p(n, A, M) ≤ nc for all n > n_{0}. This answers a question of Ljujić and Nathanson.

Original language | English (US) |
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Title of host publication | Integers |

Subtitle of host publication | Annual Volume 2013 |

Publisher | Walter de Gruyter GmbH |

Pages | 228-236 |

Number of pages | 9 |

ISBN (Electronic) | 9783110298161 |

ISBN (Print) | 9783110298116 |

DOIs | |

State | Published - Jan 1 2014 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- General Mathematics