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Permutation Totally Symmetric Self-Complementary Plane Partitions

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Abstract

Alternating sign matrices and totally symmetric self-complementary plane partitions are equinumerous sets of objects for which no explicit bijection is known. In this paper, we identify a subset of totally symmetric self-complementary plane partitions corresponding to permutations by giving a statistic-preserving bijection to permutation matrices, which are a subset of alternating sign matrices. We use this bijection to define a new partial order on permutations, and prove this new poset contains both the Tamari lattice and the Catalan distributive lattice as subposets. We also study a new partial order on totally symmetric self-complementary plane partitions arising from this perspective and show that this is a distributive lattice related to Bruhat order when restricted to permutations.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Ayyer, A., Cori, R., Gouyou-Beauchamps, D.: Monotone triangles and 312 pattern avoidance. Electron. J. Combin. 18(2), #P26 (2011)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Bettinelli, J.: A simple explicit bijection between (n, 2)-Gog and Magog trapezoids. Sém. Lothar. Combin. 75, Art. B75e (2015)

  6. Björner A., Brenti F.: Combinatorics of Coxeter Groups Grad. Texts in Math. Vol. 231. Springer, New York (2005)

    MATH  Google Scholar 

  7. Bressoud D.M.: Proofs and Confirmations. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  8. Cheballah, H., Biane, P.: Gog and Magog triangles, and the Schützenberger involution. Sém. Lothar. Combin. 66, Art. B66d (2011/12)

  9. Cheong, O.: The Ipe extensible drawing editor (Version 7). http://ipe.otfried.org/ (2015)

  10. Di Francesco P., Zinn-Justin P.: Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties. Comm. Math. Phys. 262(2), 459–487 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Di Francesco, P.: Integrable combinatorics. In: Jensen, A. (ed.) XVIIth International Congress on Mathematical Physics, pp. 29–51. World Sci. Publ., Hackensack, NJ (2014)

  12. Di Francesco P.: Totally symmetric self-complementary plane partitions and the quantum Knizhnik-Zamolodchikov equation: a conjecture. J. Stat. Mech. Theory Exp. 2006(9), P09008 (2006)

    Article  MathSciNet  Google Scholar 

  13. Doran, W.F, IV: A connection between alternating sign matrices and totally symmetric self-complementary plane partitions. J. Combin. Theory Ser. A 64(2), 289–310 (1993)

  14. Elkies N., Kuperberg G., Larsen M., Propp J.: Alternating-sign matrices and domino tilings I. J. Algebraic Combin. 1(2), 111–132 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fonseca, T., Zinn-Justin, P.: On the doubly refined enumeration of alternating sign matrices and totally symmetric self-complementary plane partitions. Electron. J. Combin. 15(1), #81, (2008)

  16. Huang S., Tamari D.: Problems of associativity: a simple proof for the lattice property of systems ordered by a semi-associative law. J. Combin. Theory Ser. A 13, 7–13 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kim, J.S.: TiKZ code for plane partitions. Available at http://www.texample.net/tikz/examples/plane-partition/

  18. Kuperberg G.: Another proof of the ASM conjecture. Internat. Math. Res. Notices 1996(3), 139–150 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lascoux, A., Schützenberger, M.-P.: Treillis et bases des groupes de Coxeter. Electron. J. Combin. 3(2), #R27 (1996)

  20. Mills W.H., Robbins D.P., Rumsey H., Rumsey H.: Alternating sign matrices and descending plane partitions. J. Combin. Theory Ser. A 34(3), 340–359 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mills W.H., Robbins D.P., Rumsey H. Jr.: Proof of the Macdonald conjecture. Invent. Math. 66(1), 73–87 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mills W.H., Robbins D.P., Rumsey H. Jr.: Self complementary totally symmetric plane partitions. J. Combin. Theory Ser. A 42(2), 277–292 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Müller-Hoissen F., Pallo J.M., Stasheff J.: Associahedra, Tamari lattices and Related Structures Progr. Math.Vol. 299. Birkhäuser Boston Inc, Boston, MA (2012)

    Book  MATH  Google Scholar 

  24. Propp, J.: The many faces of alternating-sign matrices. In: Discrete Models: Combinatorics, Computation, and Geometry, Discrete Mathematics & Theoretical Computer Science Proceedings, AA., pp. 43–58. Maison Inform. Math. Discrèt. (MIMD), Paris (2001)

  25. Robbins D., Rumsey H. Jr.: Determinants and alternating sign matrices. Adv. Math. 62(2), 169–184 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stein, W.A. et al.: Sage Mathematics Software (Version 6.5). The Sage Development Team, http://www.sagemath.org (2015)

  27. The Sage-Combinat community. Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics. http://combinat.sagemath.org (2008)

  28. Stanley R.: Enumerative Combinatorics Volume 1. Second edition, Cambridge Stud. Adv. Math. Vol. 49. Cambridge University Press, Cambridge. (2012)

    Google Scholar 

  29. Stanley R.: Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebraic Discrete Methods, 1(2), 168–184 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Striker J.: A direct bijection between descending plane partitions with no special parts and permutation matrices. Discrete Math. 311(21), 2581–2585 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Striker J., Williams N.: Promotion and rowmotion. European J. Combin. 33(8), 1919–1942 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Striker, J.: The toggle group, homomesy, and the Razumov-Stroganov correspondence. Electron. J. Combin. 22(2), #P2.57 (2015)

  33. Striker J.: A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux. Adv. Appl. Math. 46(1-4), 583–609 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zeilberger, D.: Proof of the alternating sign matrix conjecture. Electron. J. Combin. 3(2), #R13 (1996)

  35. Zinn-Justin P., Di Francesco P.: Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions and alternating sign matrices. Theoret. Math. Phys. 154(3), 331–348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, North Dakota State University, Fargo, ND, 58108-6050, USA

    Jessica Striker

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  1. Jessica Striker
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Correspondence to Jessica Striker.

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The author is supported in part by the National Security Agency grant number H98230-15-1-0041, North Dakota EPSCoR grant number IIA-1355466, and the NDSU Advance FORWARD program sponsored by the National Science Foundation HRD-0811239.

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Striker, J. Permutation Totally Symmetric Self-Complementary Plane Partitions. Ann. Comb. 22, 641–671 (2018). https://doi.org/10.1007/s00026-018-0394-0

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