Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON ROGERS-RAMANUJAN TYPE IDENTITIES FOR OVERPARTITIONS AND GENERALIZED LATTICE PATHS

  • Goyal, Megha (Department of Mathematical Sciences I. K. Gujral Punjab Technical University Jalandhar)
  • Received : 2017.01.24
  • Accepted : 2017.12.29
  • Published : 2018.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we introduce and study the lattice paths for which the horizontal step is allowed at height $h{\geq}0$, $h{\in}{\mathbb{Z}}$. By doing so these paths generalize the heavily studied weighted lattice paths that consist of horizontal steps allowed at height zero only. Six q-series identities of Rogers-Ramanujan type are studied combinatorially using these generalized lattice paths. The results are further extended by using (n + t)-color overpartitions. Finally, we will establish that there are certain equinumerous families of (n + t)-color overpartitions and the generalized lattice paths.

Keywords

References

  1. A. K. Agarwal, Lattice paths and Rogers-Ramanujan type identities, In Proceedings of the 14th annual Conf. of the Ramanujan Math. Soc., 31-39, RMS, India, 1999.
  2. A. K. Agarwal and G. E. Andrews, Hook differences and lattice paths, J. Statist. Plann. Inference 14 (1986), no. 1, 5-14. https://doi.org/10.1016/0378-3758(86)90004-2
  3. A. K. Agarwal and G. E. Andrews, Rogers-Ramanujan identities for partitions with "n copies of n", J. Combin. Theory Ser. A 45 (1987), no. 1, 40-49. https://doi.org/10.1016/0097-3165(87)90045-8
  4. A. K. Agarwal and D. M. Bressoud, Lattice paths and multiple basic hypergeometric series, Pacific J. Math. 136 (1989), no. 2, 209-228. https://doi.org/10.2140/pjm.1989.136.209
  5. A. K. Agarwal and M. Goyal, Lattice paths and Rogers identities, Open J. Discrete Math. 1 (2011), no. 2, 89-95. https://doi.org/10.4236/ojdm.2011.12011
  6. A. K. Agarwal and M. Goyal, On 3-way combinatorial identities, Proc. Indian Acad. Sci. (Math. Sci.), to appear.
  7. W. Chu and W. Zhang, Bilateral Bailey lemma and Rogers-Ramanujan identities, Adv. in Appl. Math. 42 (2009), no. 3, 358-391. https://doi.org/10.1016/j.aam.2008.07.003
  8. S. Corteel and O. Mallet, Overpartitions, lattice paths, and Rogers-Ramanujan identities, J. Combin. Theory Ser. A 114 (2007), no. 8, 1407-1437. https://doi.org/10.1016/j.jcta.2007.02.004
  9. J. Eom, G. Jeong, and J. Sohn, Three different ways to obtain the values of hyper m-ary partition functions, Bull. Korean Math. Soc. 53 (2016), no. 6, 1857-1868. https://doi.org/10.4134/BKMS.b151039
  10. M. Goyal, New combinatorial interpretations of some Rogers-Ramanujan type identities, Contrib. Discrete Math. 11 (2017), no. 2, 43-57.
  11. M. Goyal and A. K. Agarwal, Further Rogers-Ramanujan identities for n-color partitions, Util. Math. 95 (2014), 141-148.
  12. M. Goyal and A. K. Agarwal, On a new class of combinatorial identities, Ars Combin. 127 (2016), 65-77.
  13. J. Lovejoy and O. Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci. 6 (2008), no. 2, 23-36.
  14. S. Seo, Combinatorial enumeration of the regions of some linear arrangements, Bull. Korean Math. Soc. 53 (2016), no. 5, 1281-1289. https://doi.org/10.4134/BKMS.b150404