East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

ON AN INVOLUTION ON PARTITIONS WITH CRANK 0

  • Kim, Byungchan (School of Liberal Arts, Seoul National University of Science and Technology)
  • Received : 2018.07.03
  • Accepted : 2018.10.20
  • Published : 2019.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Kaavya introduce an involution on the set of partitions with crank 0 and studied the number of partitions of n which are invariant under Kaavya's involution. If a partition ${\lambda}$ with crank 0 is invariant under her involution, we say ${\lambda}$ is a self-conjugate partition with crank 0. We prove that the number of such partitions of n is equal to the number of partitions with rank 0 which are invariant under the usual partition conjugation. We also study arithmetic properties of such partitions and their q-theoretic implication.

Keywords

References

  1. G.E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, 1976; reissued: Cambridge University Press, Cambridge, 1998.
  2. G. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc. 18 (1988), 167-171. https://doi.org/10.1090/S0273-0979-1988-15637-6
  3. B.C. Berndt, Number theory in the spirit of Ramanujan, American Mathematical Society, Providence, RI, 2006.
  4. W.Y.C. Chen, K. Q. Ji, and W.J.T. Zang, Proof of the Andrews-Dyson-Rhoades conjecture on the spt-crank, Adv. Math. 270 (2015), 60-96. https://doi.org/10.1016/j.aim.2014.10.017
  5. F. Dyson, Some guesses in the theory of partitions, Eureka 8 (1944), 10-15.
  6. F. Garvan, New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7, and 11, Trans. Amer. Math. Soc. 305 (1988), 47-77. https://doi.org/10.1090/S0002-9947-1988-0920146-8
  7. S. J. Kaavya, Crank 0 partitions and the parity of the partition function, Int. J. Number Theory, 7 (2011), 793-801. https://doi.org/10.1142/S1793042111004381