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Partition identities and geometric bijections


Author: Igor Pak
Journal: Proc. Amer. Math. Soc. 132 (2004), 3457-3462
MSC (2000): Primary 05A17; Secondary 05A15, 05A19, 11P81
DOI: https://doi.org/10.1090/S0002-9939-04-07031-5
Published electronically: July 14, 2004
MathSciNet review: 2084064
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Abstract: We present a geometric framework for a class of partition identities. We show that there exists a unique bijection proving these identities, which satisfies certain linearity conditions. In particular, we show that Corteel's bijection enumerating partitions with nonnegative $r$-th differences can be obtained by our approach. Other examples and generalizations are presented.


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

  • [A1] G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, 1976. MR 58:27738
  • [A2] G. E. Andrews, A note on partitions and triangles with integer sides, Amer. Math. Monthly 86 (1979), 477-478. MR 80g:10009
  • [A3] G. E. Andrews, MacMahon's Partition analysis. II. Fundamental theorems, Ann. Combin. 4 (2000), 327-338. MR 2002g:05014
  • [APR] G. E. Andrews, P. Paule, A. Riese, MacMahon's partition analysis. III. The Omega package, European J. Combin. 22 (2001), 887-904. MR 2002h:11100
  • [BE] M. Bousquet-Mélou, K Eriksson, Lecture Hall Partitions 2, Ramanujan J. 1 (1997), 165-185. MR 99c:05016
  • [CCH] R. Canfield, S. Corteel, P. Hitczenko, Random partitions with non-negative $r$-th differences, Adv. in Appl. Math. 27 (2001), 298-317. MR 2002j:05012
  • [CS] S. Corteel, C. Savage, Partitions and Compositions defined by inequalities, Ramanujan J. (to appear).
  • [H] D. R. Hickerson, A partition identity of the Euler type, Amer. Math. Monthly 81 (1974), 627-629. MR 49:2526
  • [JWW] J. H. Jordan, Ray Walch, R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly 86 (1979), 686-689. MR 81f:05013
  • [Sl] N. J. A. Sloane, On-line Encyclopedia of Integer Sequences, available at http://www. research.att.com/ $\widetilde \,$njas/sequences. Electron. J. Combin. 1 (1994). MR 95b:05001
  • [St] R. P. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997. MR 98a:05001
  • [Z] D. Zeilberger, Sylvie Corteel's One-Line Proof of a Partition Theorem Generated by Andrews-Paule-Riese's Computer, Shalosh B. Ekhad's and Doron Zeilberger's Very Own Journal, available at: http://www.math.rutgers.edu/ $\widetilde \,$zeilberg/mamarim/mamarimhtml/ corteel.html.

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Additional Information

Igor Pak
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

DOI: https://doi.org/10.1090/S0002-9939-04-07031-5
Keywords: Partition identities, bijections, simple cones
Received by editor(s): July 24, 2002
Received by editor(s) in revised form: August 30, 2002
Published electronically: July 14, 2004
Communicated by: John R. Stembridge
Article copyright: © Copyright 2004 American Mathematical Society

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