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Dichotomy of the Addition of Natural Numbers

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Associahedra, Tamari Lattices and Related Structures

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References

  1. J.-C.Aval and X. Viennot, “The product of trees in the Loday-Ronco algebra through Catalan alternative tableaux”, Sém. Lothar. Combin. 63 (2010), Art. B63h, 8 pp.

    Google Scholar 

  2. M. Aguiar and F. Sottile, “Structure of the Loday-Ronco Hopf algebra of trees”, J. Algebra 295 (2006) 473–511.

    Article  MathSciNet  MATH  Google Scholar 

  3. Ch. Brouder, “Trees, renormalization and differential equations”, BIT 44 (2004) 425–438.

    Article  MathSciNet  MATH  Google Scholar 

  4. Ch. Brouder and A. Frabetti, “QED Hopf algebras on planar binary trees”, J. Algebra 267 (2003) 298–322.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Bruno and D. Yazaki, “The arithmetic of trees”, arxiv.org/abs/0809.4448 .

  6. E. Burgunder and M. Ronco, “Tridendriform structure on combinatorial Hopf algebras”, J. Algebra 324 (2010) 2860–2883.

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Chapoton, “Opérades différentielles graduées sur les simplexes et les permutoèdres”, Bull. Soc. Math. France 130 (2002) 233–251.

    MathSciNet  MATH  Google Scholar 

  8. S. Devadoss, “A realization of graph associahedra”, Discrete Math. 309 (2009) 27–276.

    Article  MathSciNet  Google Scholar 

  9. V. Dotsenko, “Compatible associative products and trees”, Algebra Number Theory 3 (2009) 567–586.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Ebrahimi-Fard, “Loday-type algebras and the Rota-Baxter relation”, Lett. Math. Phys. 61 (2002) 139–147.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Forcey, “Quotients of the multiplihedron as categorified associahedra”, Homology, Homotopy Appl. 10 (2008) 227–256.

    MathSciNet  MATH  Google Scholar 

  12. S. Forcey, “Convex hull realizations of the multiplihedra”, Topology Appl. 156 (2008) 326–347.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. de Fougères, “Propriétés géométriques liées aux parenthésages d’ un produit de m facteurs. Application à une loi de composition partielle”, Séminaire Dubreuil. Algèbre et théorie des nombres 18, no 2 (1964-65), exp. no 20, 1–21.

    Google Scholar 

  14. H. Gangl, A.B. Goncharov and A. Levin, “Multiple polylogarithms, polygons, trees and algebraic cycles”, in Algebraic geometry, Seattle 2005, Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc., Providence, RI, 2009, 547–593.

    Google Scholar 

  15. Ch. Hohlweg and C. Lange, “Realizations of the associahedron and cyclohedron”, Discrete Comput. Geom. 37 (2007) 517–543.

    Article  MathSciNet  MATH  Google Scholar 

  16. J.-L. Loday, “Algèbres ayant deux opérations associatives (digèbres)”, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 141–146.

    MathSciNet  MATH  Google Scholar 

  17. J.-L. Loday, “Homotopical syzygies”, in “Une dégustation topologique: Homotopy theory in the Swiss Alps”, Contemporary Mathematics no 265 (AMS) (2000), 99–127.

    Google Scholar 

  18. J.-L. Loday, “Dialgebras”, in Dialgebras and related operads, Springer Lecture Notes in Math. 1763 (2001) 7–66.

    Article  MathSciNet  Google Scholar 

  19. J.-L. Loday, “Arithmetree”, Journal of Algebra 258 (2002) 275–309.

    Article  MathSciNet  MATH  Google Scholar 

  20. J.-L. Loday, “Realization of the Stasheff polytope”, Archiv der Mathematik 83 (2004) 267–278.

    Article  MathSciNet  MATH  Google Scholar 

  21. J.-L. Loday, “The diagonal of the Stasheff polytope”, in Higher structures in geometry and physics, Progr. Math. 287, Birkhäuser, Basel, 2011, 269–292.

    Google Scholar 

  22. J.-L. Loday, “Geometric diagonals for the Stasheff associahedron and products of A-infinity algebras”, preprint 2011, in preparation.

    Google Scholar 

  23. J.-L. Loday; M.O. Ronco, “Hopf algebra of the planar binary trees”, Adv. Math. 139 (1998) 293–309.

    Article  MathSciNet  MATH  Google Scholar 

  24. J.-L. Loday and M.O. Ronco, “Order structure and the algebra of permutations and of planar binary trees”, J. Alg. Comb. 15 (2002) 253–270.

    Article  MathSciNet  MATH  Google Scholar 

  25. J.-C. Novelli and J.-Y. Thibon, “Hopf algebras and dendriform structures arising from parking functions”, Fund. Math. 193 (2007) 189–241.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Postnikov, “Permutohedra, associahedra, and beyond”, Int. Math. Res. Not. 2009, no. 6, 1026–1106.

    Google Scholar 

  27. V. Pilaud and F. Santos, “Multitriangulations as complexes of star polygons”, Discrete Comput. Geom. 41 (2009) 284–317.

    Article  MathSciNet  MATH  Google Scholar 

  28. M.O. Ronco, “Primitive elements in a free dendriform algebra”, in New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math. 267, Amer. Math. Soc., Providence, RI, 2000, 245–263.

    Google Scholar 

  29. S. Saneblidze and R. Umble, “Diagonals on the permutahedra, multiplihedra and associahedra”, Homology Homotopy Appl. 6 (2004) 363–411.

    MathSciNet  MATH  Google Scholar 

  30. J. Stasheff, “Homotopy associativity of H-spaces. I, II”, Trans. Amer. Math. Soc. 108 (1963) 275-292; ibid. 108 (1963) 293–312.

    Google Scholar 

  31. J. Stasheff, “From operads to “physically” inspired theories”, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math. 202, Amer. Math. Soc., Providence, RI, 1997, 53–81.

    Google Scholar 

  32. J. Stasheff, “How I ’ met’ Dov Tamari”, in Tamari Memorial Festschrift.

    Google Scholar 

  33. D. Tamari, “Monoides préordonnés et chaînes de Malcev”, Doctorat ès-Sciences Mathématiques Thèse de Mathématiques, Université de Paris (1951).

    Google Scholar 

  34. D. Tamari, “Monoides préordonnés et chaînes de Malcev”, Bull. Soc. Math. France 82 (1954) 53–96.

    MathSciNet  MATH  Google Scholar 

  35. B. Vallette, “Manin products, Koszul duality, Loday algebras and Deligne conjecture”, J. Reine Angew. Math. 620 (2008) 105–164.

    MathSciNet  MATH  Google Scholar 

  36. D. Yau, “Gerstenhaber structure and Deligne’ s conjecture for Loday algebras”, J. Pure Appl. Algebra 209 (2007) 739–752.

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Jean-Louis Loday .

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Loday, JL. (2012). Dichotomy of the Addition of Natural Numbers. In: Müller-Hoissen, F., Pallo, J., Stasheff, J. (eds) Associahedra, Tamari Lattices and Related Structures. Progress in Mathematics, vol 299. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0405-9_4

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