Research ArticleOpen AccessGoogle Scholar indexed
Lattice Paths and Rogers Identities
- 1
- 2
Open Journal of Discrete Mathematics·Volume 01 (2011)·Pages 89–95·Published 5 July 2011·DOI10.4236/ojdm.2011.12011
Copy link · social · email
Abstract
Recently we interpreted five q-series identities of Rogers combinatorially by using partitions with “<i>n</i> +<i>t</i> cop-ies of <i>n</i>” of Agarwal and Andrews (J. Combin. Theory Ser.A, 45(1987), No.1, 40-49). In this paper we use lattice paths of Agarwal and Bressoud (Pacific J. Math. 136(2) (1989), 209-228) to provide new combinatorial interpretations of the same identities. This results in five new 3-way combinatorial identities.
KeywordsLattice PathsColored PartitionsGenerating FunctionsCombinatorial Interpretations
- A. K. Agarwal and G. E. Andrews, “Rogers-Ramanujan Identities for Partitions with ‘ Copies of ’,” Journal of Combinatorial Theory, Vol. 45, No. 1, 1987, pp. 40-49. doi:10.1016/0097-3165(87)90045-8
- A. K. Agarwal and D. M. Bressoud, “Lattice Paths and Multiple Basic Hypergeometric Series,” Pacific Journal of Mathematics, Vol. 136, No. 2, 1989, pp. 209-228.
- L. J. Slater, “Further Identities of the Rogers-Ramanujan Type,” Proceedings of the London Mathematical Society, Vol. s2-54, No. 1, 1952, pp. 147-167. doi:10.1112/plms/s2-54.2.147
- W. G. Connor, “Partition Theorems Related to some Id- entities of Rogers and Watson,” Transactions of the Ame- rican Mathematical Society, Vol. 214, 1975, pp. 95-111. doi:10.1090/S0002-9947-1975-0414480-9
- M. V. Subbarao, “Some Rogers-Ramanujan Type Partition Theorems,” Pacific Journal of Mathematics, Vol. 120, 1985, pp. 431-435.
- M. V. Subbarao and A. K. Agarwal, “Further Theorems of the Rogers-Ramanujan Type,” Canadian Mathematical Bulletin, Vol. 31, No. 2, 1988, pp. 210-214. doi:10.4153/CMB-1988-032-3
- A. K. Agarwal and G. E. Andrews, “Hook Differences and Lattice Paths,” Journal of Statistical Planning and Inference, Vol. 14, No. 1, 1986, pp. 5-14. doi:10.1016/0378-3758(86)90004-2
- A. K. Agarwal, “Partitions with ‘ copies of ’, Lecture Notes in Math.,” Proceedings of the Colloque de Combinatoire énumérative, Université du Québec à Montréal, Berlin, May 28-June 1, 1985, No. 1234, pp. 1-4.
- A. K. Agarwal, “Rogers-Ramanujan Identities for n- Color Partitions,” Journal of Number Theory, Vol. 28, No. 3, 1988, pp. 299-305. doi:10.1016/0022-314X(88)90045-5
- A. K. Agarwal, “New Combinatorial Interpretations of Two Analytic Identities,” Proceedings of the American Mathematical Society, Vol. 107, No. 2, 1989, pp. 561-567. doi:10.1090/S0002-9939-1989-0979216-7
- A. K. Agarwal, “New Classes of Infinite 3-Way Partition Identities,” ARS Combinatoria, Vol. 44, 1996, pp. 33-54.
- A .K. Agarwal and G. E. Andrews, “Rogers-Ramanujan Identities for Partitions with ‘ Copies of ’,” Journal of Combinatorial Theory, Series A, Vol. 45, No. 1, 1987, pp. 40-49. doi:10.1016/0097-3165(87)90045-8
- M. Goyal and A. K. Agarwal, “Further Rogers-Rama- nujan Identities for n-Color Partitions,” Utilitas Methematica, Winnipeg. (In Press)
- L. J. Rogers, “Second Memoir on the Expansion of Certain Infinite Products,” Proceedings of the London Ma- thematical Society, Vol. s1-25, No. 1, 1894, pp. 318-343. doi:10.1112/plms/s1-25.1.318