Abstract
Letp j(m, n) be the number of partitions of (m, n) into at mostj parts. We prove Landman et al.'s conjecture: for allj andn, p j(x, 2n−x) is a maximum whenx-n. More generally we prove that for all positive integersm, n andj, p j(n, m)=pj(m, n)≥pj(m−1, n+1) ifm≤n.
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References
Andrews, G.E.: The Theory of Partitions, Reading, Massachusetts: Addison-Wesley Publishing Company, 1976
Landmam, B.M., Brown, E.A., Portier, F.J.: Partitions of bi-partite numbers into at mostj parts, Graph. Comb.8, 65–73 (1992)
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Kim, J.K., Hahn, S.G. Partitions of bipartite numbers. Graphs and Combinatorics 13, 73–78 (1997). https://doi.org/10.1007/BF01202238
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DOI: https://doi.org/10.1007/BF01202238