Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE NUMBER OF EQUIVALENCE CLASSES OF BI-PARTITIONS ARISING FROM THE COLOR CHANGE

  • Byungchan Kim (School of Natural Sciences Seoul National University of Science and Technology)
  • Received : 2023.08.18
  • Accepted : 2024.03.04
  • Published : 2024.04.30
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Abstract

We introduce a new class of bi-partition function ck(n), which counts the number of bi-color partitions of n in which the second color only appears at the parts that are multiples of k. We consider two partitions to be the same if they can be obtained by switching the color of parts that are congruent to zero modulo k. We show that the generating function for ck(n) involves the partial theta function and obtain the following congruences: c2(27n + 26) ≡ 0 (mod 3) and c3(4n + 2) ≡ 0 (mod 2).

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Acknowledgement

The author is grateful to the anonymous referee for the valuable suggestions. In particular, the referee notices a connection between the generating function for p2k,k(n) in Andrews and Newman [3] and the generating function for ck(n).

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