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ON PARTITION CONGRUENCES FOR OVERCUBIC PARTITION PAIRS

  • Kim, Byung-Chan (School of Liberal Arts Seoul National University of Science and Technology)
  • Received : 2011.04.26
  • Published : 2012.07.31

Abstract

In this note, we investigate partition congruences for a certain type of partition function, which is named as the overcubic partition pairs in light of the literature. Let $\bar{cp}(n)$ be the number of overcubic partition pairs. Then we will prove the following congruences: $$\bar{cp}(8n+7){\equiv}0(mod\;64)\;and\;\bar{cp}(9m+3){\equiv}0(mod\;3)$$.

References

  1. G. E. Andrews and F. G. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167-171. https://doi.org/10.1090/S0273-0979-1988-15637-6
  2. B. C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, Providence, RI, 2006.
  3. H.-C. Chan, Ramanujan's cubic continued fraction and an analog of his "most beautiful identity", Int. J. Number Theory 6 (2010), no. 3, 673-680. https://doi.org/10.1142/S1793042110003150
  4. H.-C. Chan, Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function, Int. J. Number Theory 6 (2010), no. 4, 819-834. https://doi.org/10.1142/S1793042110003241
  5. H.-C. Chan, Distribution of a certain partition function modulo powers of primes, Acta Math. Sin. (Engl. Ser.) 27 (2011), 625-634.
  6. B. Gordon and K. Hughes, Ramanujan congruences for q(n), Analytic number theory (Philadelphia, Pa., 1980), 333-359, Lecture Notes in Math., 899, Springer, Berlin-New York, 1981. https://doi.org/10.1007/BFb0096473
  7. B. Kim, The overcubic partition function mod 3, Proceedings of Ramanujan Rediscovered 2009: A Conference in Memory of K. Venkatachaliengar on the Centenary of His Birth, Lecture Note Series of the Ramanujan Mathematical Society 14 (2010), 157-163.
  8. B. Kim, Overpartition pairs modulo powers of 2, Discrete Math. 311 (2011), no. 10-11, 835-840. https://doi.org/10.1016/j.disc.2011.02.002
  9. G. Ligozat, Courbes modulaires de genre 1, Bull. Soc. Math. France, Mem. 43. Supplement au Bull. Soc. Math. France Tome 103, no. 3. Societe Mathematique de France, Paris, 1975. 80 pp.
  10. K. Mahlburg, The overpartition function modulo small powers of 2, Discrete Math. 286 (2004), no. 3, 263-267. https://doi.org/10.1016/j.disc.2004.03.014
  11. M. Newman, Construction and application of a class of modular functions II, Proc. London Math. Soc. (3) 9 (1959), 373-387. https://doi.org/10.1112/plms/s3-9.3.373
  12. K. Ono, Web of Modularity: arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics, 102. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2004.
  13. H. Zhao and Z. Zhong, Ramanujan type congruences for a partition function, The Electronic Lournal of Combinatorics 18 (2011), P. 58.

Cited by

  1. Congruences modulo 27 for cubic partition pairs vol.171, 2017, https://doi.org/10.1016/j.jnt.2016.07.012
  2. New congruences for overcubic partition pairs vol.10, pp.4, 2017, https://doi.org/10.1515/tmj-2017-0050