### Abstract

We show that the number of partitions of n with alternating sum κ such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's partition theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a generalization of Euler's partition theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of n with alternating sum κ such that the multiplicity of each even part is bounded by 2m + 1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is also bounded by 2m+1. We provide a combinatorial proof as well.

Original language | English (US) |
---|---|

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Electronic Journal of Combinatorics |

Volume | 19 |

Issue number | 3 |

State | Published - Oct 4 2012 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*19*(3), 1-15.

}

*Electronic Journal of Combinatorics*, vol. 19, no. 3, pp. 1-15.

**Euler's partition theorem with upper bounds on multiplicities.** / Chen, William Y C; Yee, Ae Ja; Zhu, Albert J W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Euler's partition theorem with upper bounds on multiplicities

AU - Chen, William Y C

AU - Yee, Ae Ja

AU - Zhu, Albert J W

PY - 2012/10/4

Y1 - 2012/10/4

N2 - We show that the number of partitions of n with alternating sum κ such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's partition theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a generalization of Euler's partition theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of n with alternating sum κ such that the multiplicity of each even part is bounded by 2m + 1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is also bounded by 2m+1. We provide a combinatorial proof as well.

AB - We show that the number of partitions of n with alternating sum κ such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's partition theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a generalization of Euler's partition theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of n with alternating sum κ such that the multiplicity of each even part is bounded by 2m + 1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is also bounded by 2m+1. We provide a combinatorial proof as well.

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UR - http://www.scopus.com/inward/citedby.url?scp=84867440248&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84867440248

VL - 19

SP - 1

EP - 15

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 3

ER -