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Restricted k-color partitions

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Abstract

We generalize overpartitions to (k, j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems: generating function dissections, modular forms, bijections, and other combinatorial maps. In the process of proving certain congruences, we find results of independent interest on the number of partitions with exactly 2 sizes of part in several arithmetic progressions. We find connections to divisor sums, the Han/Nekrasov–Okounkov hook length formula and a possible approach to finitization, and other topics, suggesting that a rich mine of results is available. We pose several immediate questions and conjectures.

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Acknowledgments

For catching several typographical errors and suggesting numerous references and improvements to presentation in this paper, the author cordially thanks the anonymous referees for their work.

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  1. Michigan Tech University Houghton, Houghton, MI, 49931, USA

    William J. Keith

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  1. William J. Keith
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Correspondence to William J. Keith.

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Keith, W.J. Restricted k-color partitions. Ramanujan J 40, 71–92 (2016). https://doi.org/10.1007/s11139-015-9704-x

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