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
MathSciNet review:
3611316
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.
- [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 123484, https://doi.org/10.2307/2372962
- [10]
Omar Foda and Trevor Welsh, Cylindric partitions,
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. 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 272654
- [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. 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 ̂𝑠𝑙_{𝑛} crystals, with applications to cylindric plane partitions, Int. Math. Res. Not. IMRN 2 (2008), Art. ID rnm143, 40. MR 2418856
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