**Weighted forms of Euler's theorem** |

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**Authors:** | William YC Chen |

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**Institution:** | Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR China |

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**Abstract:** | In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's “lost” notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan's identities and Euler's theorem. Our combinatorial formulations of Ramanujan's identities rely on the notion of rooted partitions. Pak's iterated Dyson's map and Sylvester's fish-hook bijection are the main ingredients in the weighted forms of Euler's theorem. |

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**Keywords:** | Partition Rooted partition Euler's theorem Ramanujan's identities Pak's iterated Dyson's map Sylvester's fish-hook bijection |

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