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Gog and Magog Triangles

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Computation and Combinatorics in Dynamics, Stochastics and Control (Abelsymposium 2016)

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Abstract

We survey the problem of finding an explicit bijection between Gog and Magog triangles, a combinatorial problem which has been open since the 1980s. We give some of the ideas behind a recent approach to this question and also prove some properties of the distribution of inversions and coinversions in Gog triangles.

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Notes

  1. 1.

    In the first version of this paper the statement of this proposition was incorrect. I would like to thank the referees for pointing out the mistake.

References

  1. Andrews G.E.: The Theory of Partitions. Encyclopedia of Mathematics and its Applications, vol. 2. Addison-Wesley Publishing Co., Reading/London/Amsterdam (1976)

    Google Scholar 

  2. Andrews G.E.: Plane partitions. V. The TSSCPP conjecture. J. Combin. Theor. Ser. A 66(1), 28–39 (1994)

    Article  MathSciNet  Google Scholar 

  3. Ayyer, A., Romik, D.: New enumeration formulas for alternating sign matrices and square ice partition functions. Adv. Math. 235, 161–186 (2013)

    Article  MathSciNet  Google Scholar 

  4. Baxter, R.: Exactly Solved Models in Statistical Mechanics. Academic, London (1982)

    MATH  Google Scholar 

  5. Behrend R.E.: Multiply-refined enumeration of alternating sign matrices. Adv. Math. 245, 439–499 (2013)

    Article  MathSciNet  Google Scholar 

  6. Behrend, R.E., Di Francesco, P., Zinn-Justin, P.: On the weighted enumeration of alternating sign matrices and descending plane partitions. J. Combin. Theor. Ser. A 119(2), 331–363 (2012)

    Article  MathSciNet  Google Scholar 

  7. Berenstein, A.D., Kirillov, A.N.: Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux. St. Petersburg Math. J. 7(1), 77–127 (1996)

    Google Scholar 

  8. Bettinelli, J.: A simple explicit bijection between (n,2)-Gog and Magog trapezoids. Lotharingien Séminaire Lotharingien de Combinatoire 75, 1–9 (2016). Article B75e

    Google Scholar 

  9. Biane, P., Cheballah, H.: Gog and Magog triangles and the Schützenberger involution. Séminaire Lotharingien de Combinatoire, B66d (2012)

    Google Scholar 

  10. Biane, P., Cheballah, H.: Gog and GOGAm pentagons. J. Combin. Theor. Ser. A 138, 133–154 (2016) JCTA

    Article  MathSciNet  Google Scholar 

  11. Bressoud D.M.: Proofs and Confirmations, the Story of the Alternating Sign Matrix Conjecture. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  12. Cantini, L., Sportiello, A.: Proof of the Razumov-Stroganoff conjecture. J. Combin. Theor. Ser. A 118(5), 1549–1574 (2011)

    Article  Google Scholar 

  13. Fischer I.: A new proof of the refined alternating sign matrix theorem. J. Combin. Theor. Ser. A 114(2), 253–264 (2007)

    Article  MathSciNet  Google Scholar 

  14. Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. Appl. Math. 28, 119–144 (2002)

    Article  MathSciNet  Google Scholar 

  15. Fulton, W.: Young Tableaux. London Mathematical Society, Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  16. Izergin, A.G.: Partition function of a six vertex model in a finite volume. Soviet Phys. Dokl. 32, 878–879 (1987)

    MATH  Google Scholar 

  17. Kuperberg G.: Another proof of the alternating sign matrix conjecture. Int. Math. Res. Not. 1996, 139–150 (1996)

    Article  MathSciNet  Google Scholar 

  18. Krattenthaler, C.: A Gog-Magog conjecture. http://www.mat.univie.ac.at/~kratt/artikel/magog.html

  19. Lascoux, A., Schützenberger, M.P.: Treillis et bases des groupes de Coxeter. Electron. J. Combin. 3(2), 27, 35pp (1996)

    Google Scholar 

  20. van Leeuwen, M.A.: Flag varieties and interpretations of Young tableaux algorithms. J. Algebra 224, 397–426 (2000)

    Article  MathSciNet  Google Scholar 

  21. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. With contributions by Zelevinsky, A. Oxford Mathematical Monographs. Oxford Science Publications/The Clarendon Press/Oxford University Press, New York (1995)

    Google Scholar 

  22. Mills, W.H., Robbins, D.P., Rumsey, H.: Self complementary totally symmetric plane partitions. J. Combin. Theor. Ser. A 42, 277–292 (1986)

    Article  MathSciNet  Google Scholar 

  23. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part III. Springer, New York (2012)

    Google Scholar 

  24. Robbins, D.P., Rumsey, H.: Determinants and alternating sign matrices. Adv. Math. 62, 169–184 (1986)

    Article  MathSciNet  Google Scholar 

  25. Razumov, A.V., Stroganov, Y.G.: Combinatorial nature of ground state vector of O(1) loop model. Theor. Math. Phys. 138, 333–337 (2004)

    Article  MathSciNet  Google Scholar 

  26. Stembridge, J.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83, 96–131 (1990)

    Article  MathSciNet  Google Scholar 

  27. Striker, J.: A direct bijection between permutations and a subclass of totally symmetric self-complementary plane partitions. In: 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), Paris, pp. 803–812. Discrete Mathematics and Theoretical Computer Science Proceedings, AS

    Google Scholar 

  28. Zeilberger, D.: Proof of the alternating sign matrix conjecture. Electron. J. Combin. 3, R13 (1996)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut Gaspard-Monge, Université Paris-Est Marne-la-Vallée, Marne-la-Vallée cedex 2, France

    Philippe Biane

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  1. Philippe Biane
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Correspondence to Philippe Biane .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Biane, P. (2018). Gog and Magog Triangles. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_4

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