Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Rogers-Ramanujan identities and the Robinson-Schensted-Knuth correspondence


Author: Sylvie Corteel
Journal: Proc. Amer. Math. Soc. 145 (2017), 2011-2022
MSC (2010): Primary 05A15, 05A17, 05A30, 05E10, 11P81
DOI: https://doi.org/10.1090/proc/13373
Published electronically: December 9, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives a simple combinatorial proof of the second
Rogers-Ramanujan identity by using cylindric plane partitions and the
Robinson-Schensted-Knuth algorithm.


References [Enhancements On Off] (What's this?)

  • [1] George E. Andrews, On the general Rogers-Ramanujan theorem, American Mathematical Society, Providence, R.I., 1974. Memiors of the American Mathematical Society, No. 152. MR 0364082
  • [2] Alexei Borodin, Periodic Schur process and cylindric partitions, Duke Math. J. 140 (2007), no. 3, 391-468. MR 2362241, https://doi.org/10.1215/S0012-7094-07-14031-6
  • [3] Cilanne Boulet and Igor Pak, A combinatorial proof of the Rogers-Ramanujan and Schur identities, J. Combin. Theory Ser. A 113 (2006), no. 6, 1019-1030. MR 2244131, https://doi.org/10.1016/j.jcta.2005.09.007
  • [4] Jérémie Bouttier, Guillaume Chapuy, and Sylvie Corteel, From Aztec diamonds to pyramids: steep tilings, Trans. Amer. Math. Soc., to appear (2016).
  • [5] David M. Bressoud, Lattice paths and the Rogers-Ramanujan identities, Number theory, Madras 1987, Lecture Notes in Math., vol. 1395, Springer, Berlin, 1989, pp. 140-172. MR 1019330, https://doi.org/10.1007/BFb0086403
  • [6] David M. Bressoud and Doron Zeilberger, A short Rogers-Ramanujan bijection, Discrete Math. 38 (1982), no. 2-3, 313-315. MR 676546, https://doi.org/10.1016/0012-365X(82)90298-9
  • [7] Sylvie Corteel, Cyrille Savelief, and Mirjana Vuletić, Plane overpartitions and cylindric partitions, J. Combin. Theory Ser. A 118 (2011), no. 4, 1239-1269. MR 2755080, https://doi.org/10.1016/j.jcta.2010.12.001
  • [8] Ira M. Gessel and C. Krattenthaler, Cylindric partitions, Trans. Amer. Math. Soc. 349 (1997), no. 2, 429-479. MR 1389777, https://doi.org/10.1090/S0002-9947-97-01791-1
  • [9] Basil Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math. 83 (1961), 393-399. MR 0123484
  • [10] Omar Foda and Trevor Welsh, Cylindric partitions, $ W_r$ characters and the Andrews-Gordon-Bressoud identities, 2015, arXiv:1510.02213.
  • [11] Sergey Fomin, Schensted algorithms for dual graded graphs, J. Algebraic Combin. 4 (1995), no. 1, 5-45. MR 1314558, https://doi.org/10.1023/A:1022404807578
  • [12] Sergey Fomin, Schur operators and Knuth correspondences, J. Combin. Theory Ser. A 72 (1995), no. 2, 277-292. MR 1357774, https://doi.org/10.1016/0097-3165(95)90065-9
  • [13] A. M. Garsia and S. C. Milne, A Rogers-Ramanujan bijection, J. Combin. Theory Ser. A 31 (1981), no. 3, 289-339. MR 635372, https://doi.org/10.1016/0097-3165(81)90062-5
  • [14] Sam Hopkins, RSK via local transformations. An expository article based on presentations of Alex Postnikov. http://web.mit.edu/shopkins/research.html
  • [15] Andrei Okounkov and Nikolai Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), no. 3, 581-603 (electronic). MR 1969205, https://doi.org/10.1090/S0894-0347-03-00425-9
  • [16] Donald E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709-727. MR 0272654
  • [17] C. Krattenthaler, Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes, Adv. in Appl. Math. 37 (2006), no. 3, 404-431. MR 2261181, https://doi.org/10.1016/j.aam.2005.12.006
  • [18] Robin Langer, Enumeration of cylindric plane partitions, FPSAC 2012, DMTCS proc. AR, 2012, 793-804.
  • [19] Robin Langer, Enumeration of Cylindric Plane Partitions - Part II, arXiv:1209.1807.
  • [20] Igor Pak, Partition bijections, a survey, Ramanujan J. 12 (2006), no. 1, 5-75. MR 2267263, https://doi.org/10.1007/s11139-006-9576-1
  • [21] Igor Pak, Hook length formula and geometric combinatorics, Sém. Lothar. Combin. 46 (2001/02), Art. B46f, 13 pp. (electronic). MR 1877632
  • [22] L.J. Rogers and Srinivasa Ramanujan, Proof of certain identities in combinatorial analysis, Camb. Phil. Soc. Proc., Vol 19, 1919, pp. 211-216.
  • [23] Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282
  • [24] Peter Tingley, Three combinatorial models for $ \widehat {\rm sl}_n$ crystals, with applications to cylindric plane partitions, Int. Math. Res. Not. IMRN 2 (2008), Art. ID rnm143, 40. MR 2418856

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05A15, 05A17, 05A30, 05E10, 11P81

Retrieve articles in all journals with MSC (2010): 05A15, 05A17, 05A30, 05E10, 11P81


Additional Information

Sylvie Corteel
Affiliation: IRIF, CNRS et Université Paris Diderot, Case 7014, 75251 Paris Cedex 13, France
Email: corteel@liafa.univ-paris-diderot.fr

DOI: https://doi.org/10.1090/proc/13373
Received by editor(s): November 3, 2015
Received by editor(s) in revised form: May 17, 2016, July 6, 2016, and July 13, 2016
Published electronically: December 9, 2016
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society