Gog and Magog Triangles

  • Philippe BianeEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


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.


  1. 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. 2.
    Andrews G.E.: Plane partitions. V. The TSSCPP conjecture. J. Combin. Theor. Ser. A 66(1), 28–39 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ayyer, A., Romik, D.: New enumeration formulas for alternating sign matrices and square ice partition functions. Adv. Math. 235, 161–186 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baxter, R.: Exactly Solved Models in Statistical Mechanics. Academic, London (1982)zbMATHGoogle Scholar
  5. 5.
    Behrend R.E.: Multiply-refined enumeration of alternating sign matrices. Adv. Math. 245, 439–499 (2013)MathSciNetCrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 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. 8.
    Bettinelli, J.: A simple explicit bijection between (n,2)-Gog and Magog trapezoids. Lotharingien Séminaire Lotharingien de Combinatoire 75, 1–9 (2016). Article B75eGoogle Scholar
  9. 9.
    Biane, P., Cheballah, H.: Gog and Magog triangles and the Schützenberger involution. Séminaire Lotharingien de Combinatoire, B66d (2012)Google Scholar
  10. 10.
    Biane, P., Cheballah, H.: Gog and GOGAm pentagons. J. Combin. Theor. Ser. A 138, 133–154 (2016) JCTAMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bressoud D.M.: Proofs and Confirmations, the Story of the Alternating Sign Matrix Conjecture. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  12. 12.
    Cantini, L., Sportiello, A.: Proof of the Razumov-Stroganoff conjecture. J. Combin. Theor. Ser. A 118(5), 1549–1574 (2011)CrossRefGoogle Scholar
  13. 13.
    Fischer I.: A new proof of the refined alternating sign matrix theorem. J. Combin. Theor. Ser. A 114(2), 253–264 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. Appl. Math. 28, 119–144 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fulton, W.: Young Tableaux. London Mathematical Society, Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)Google Scholar
  16. 16.
    Izergin, A.G.: Partition function of a six vertex model in a finite volume. Soviet Phys. Dokl. 32, 878–879 (1987)zbMATHGoogle Scholar
  17. 17.
    Kuperberg G.: Another proof of the alternating sign matrix conjecture. Int. Math. Res. Not. 1996, 139–150 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Krattenthaler, C.: A Gog-Magog conjecture.
  19. 19.
    Lascoux, A., Schützenberger, M.P.: Treillis et bases des groupes de Coxeter. Electron. J. Combin. 3(2), 27, 35pp (1996)Google Scholar
  20. 20.
    van Leeuwen, M.A.: Flag varieties and interpretations of Young tableaux algorithms. J. Algebra 224, 397–426 (2000)MathSciNetCrossRefGoogle Scholar
  21. 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. 22.
    Mills, W.H., Robbins, D.P., Rumsey, H.: Self complementary totally symmetric plane partitions. J. Combin. Theor. Ser. A 42, 277–292 (1986)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part III. Springer, New York (2012)Google Scholar
  24. 24.
    Robbins, D.P., Rumsey, H.: Determinants and alternating sign matrices. Adv. Math. 62, 169–184 (1986)MathSciNetCrossRefGoogle Scholar
  25. 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)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Stembridge, J.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83, 96–131 (1990)MathSciNetCrossRefGoogle Scholar
  27. 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, ASGoogle Scholar
  28. 28.
    Zeilberger, D.: Proof of the alternating sign matrix conjecture. Electron. J. Combin. 3, R13 (1996)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institut Gaspard-MongeUniversité Paris-Est Marne-la-ValléeMarne-la-Vallée cedex 2France

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