Name
Abstract | ||
|---|---|---|
We show that the number of partitions of n with alternating sum k 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 k 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. |
Year | DOI | Venue |
|---|---|---|
2012 | null | ELECTRONIC JOURNAL OF COMBINATORICS |
Keywords | Field | DocType |
partition,Euler's partition theorem,Sylvester's bijection | Discrete mathematics,Combinatorics,Bijection,Upper and lower bounds,Multiplicity (mathematics),Euler's formula,Combinatorial proof,Partition (number theory),Mathematics,Bounded function | Journal |
Volume | Issue | ISSN |
19.0 | 3.0 | 1077-8926 |
Citations | PageRank | References |
0 | 0.34 | 2 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| William Y. C. Chen | 1 | 341 | 66.51 |
| Ae Ja Yee | 2 | 44 | 12.38 |
| Albert J. W. Zhu | 3 | 3 | 0.87 |