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Applications and Generalizations

  • Samuele Giraudo
Chapter

Abstract

This last chapter is devoted to review some applications of the theory of operads for enumerative prospects. To this aim, we present formal power series on operads, generalizing usual generating series. We also provide an overview on enrichments of operads: colored operads, cyclic operads, symmetric operads, and pros.

References

  1. [BG16]
    J.-P. Bultel, S. Giraudo, Combinatorial Hopf algebras from PROs. J. Algebraic Combin. 44(2), 455–493 (2016)MathSciNetCrossRefGoogle Scholar
  2. [Boz01]
    S. Bozapalidis, Context-free series on trees. Inf. Comput. 169(2), 186–229 (2001)MathSciNetCrossRefGoogle Scholar
  3. [BR82]
    J. Berstel, C. Reutenauer, Recognizable formal power series on trees. Theor. Comput. Sci. 18(2), 115–148 (1982)MathSciNetCrossRefGoogle Scholar
  4. [BR10]
    J. Berstel, C. Reutenauer, Noncommutative Rational Series with Applications, vol. 137 (Cambridge University Press, Cambridge, 2010), pp. xiv+248Google Scholar
  5. [BV73]
    J.M. Boardman, R.M. Vogt, Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, vol. 347 (Springer, Berlin, 1973)CrossRefGoogle Scholar
  6. [CDG+07]
    H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, C. Löding, D. Lugiez, S. Tison, M. Tommasi, Tree Automata Techniques and Applications (2007). http://www.grappa.univ-lille3.fr/tata
  7. [Cha02]
    F. Chapoton, Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces. J. Pure Appl. Algebra 168(1), 1–18 (2002)MathSciNetCrossRefGoogle Scholar
  8. [Cha08]
    F. Chapoton, Operads and algebraic combinatorics of trees. Sém. Lothar. Combin. B58c, 27 (2008)Google Scholar
  9. [Cha09]
    F. Chapoton, A rooted-trees q-series lifting a one-parameter family of Lie idempotents. Algebra & Number Theory 3(6), 611–636 (2009)MathSciNetCrossRefGoogle Scholar
  10. [CO17]
    P.-L. Curien, J. Obradović, A formal language for cyclic operads. Higher Struct. 1(1), 2255 (2017)Google Scholar
  11. [Cor18]
    C. Cordero, Enumerative combinatorics of prographs, in Formal Power Series and Algebraic Combinatorics (2018)Google Scholar
  12. [Eil74]
    S. Eilenberg, Automata, Languages, and Machines. Vol. A. Pure and Applied Mathematics, vol. 58. (Academic Press, New York, 1974), pp. xvi+451Google Scholar
  13. [Fra08]
    A. Frabetti, Groups of tree-expanded series. J. Algebra 319(1), 377–413 (2008)MathSciNetCrossRefGoogle Scholar
  14. [Gir16a]
    S. Giraudo, Colored operads, series on colored operads, and combinatorial generating systems (2016). arXiv:1605.04697v1
  15. [GK95]
    E. Getzler, M.M. Kapranov, Cyclic operads and cyclic homology, in Geometry, Topology, & Physics. Conference Proceedings and Lecture Notes in Geometry and Topology, vol. IV (International Press, Cambridge, 1995), pp. 167–201Google Scholar
  16. [Har78]
    M.A. Harrison, Introduction to Formal Language Theory (Addison-Wesley Publishing Co., Reading, 1978), pp. xiv+594Google Scholar
  17. [HMU06]
    J.E. Hopcroft, R. Matwani, J.D. Ullman, Introduction to Automata Theory, Languages, and Computation, 3rd edn. (Pearson, London, 2006)Google Scholar
  18. [KP15]
    A. Khoroshkin, D. Piontkovski, On generating series of finitely presented operads. J. Algebra 426, 377–429 (2015)MathSciNetCrossRefGoogle Scholar
  19. [Laf03]
    Y. Lafont, Towards an algebraic theory of Boolean circuits. J. Pure Appl. Algebra 184(2-3), 257–310 (2003)MathSciNetCrossRefGoogle Scholar
  20. [Laf11]
    Y. Lafont, Diagram rewriting and operads. Sémin. Congr. 26, 163–179 (2011)MathSciNetzbMATHGoogle Scholar
  21. [Lei04]
    T. Leinster, Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series, vol. 298 (Cambridge University Press, Cambridge, 2004), pp. xiv+433Google Scholar
  22. [LLMN18]
    É. Laugerotte, J.-G. Luque, L. Mignot, F. Nicart, Multilinear representations of free PROs (2018). arXiv:1803.00228v1
  23. [LN13]
    J.-L. Loday, N.M. Nikolov, Operadic construction of the renormalization group, in Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics and Statistics, vol. 36 (Springer, Tokyo, 2013), pp. 191–211CrossRefGoogle Scholar
  24. [Lod01]
    J.-L. Loday, Dialgebras, in Dialgebras and Related Operads. Lecture Notes in Mathematics, vol. 1763 (Springer, Berlin, 2001), pp. 7–66Google Scholar
  25. [Lod06]
    J.-L. Loday, Completing the operadic butterfly. Georgian Math. J. 13(4), 741–749 (2006)MathSciNetzbMATHGoogle Scholar
  26. [LV12]
    J.-L. Loday, B. Vallette, Algebraic Operads. Grundlehren der mathematischen Wissenschaften, vol. 346 (Springer, Heidelberg, 2012), pp. xxiv+634CrossRefGoogle Scholar
  27. [Mar08]
    M. Markl, Operads and PROPs, in Handbook of Algebra, vol. 5 (Elsevier/North-Holland, Amsterdam, 2008), pp. 87–140zbMATHGoogle Scholar
  28. [Mén15]
    M.A. Méndez, Set Operads in Combinatorics and Computer Science. SpringerBriefs in Mathematics (Springer, Cham, 2015), pp. xvi+129Google Scholar
  29. [ML65]
    S.M. Lane, Categorical algebra. Bull. Am. Math. Soc. 71, 40–106 (1965)MathSciNetCrossRefGoogle Scholar
  30. [Sak09]
    J. Sakarovitch, Elements of Automata Theory (Cambridge University Press, Cambridge, 2009)CrossRefGoogle Scholar
  31. [SS78]
    A. Salomaa, M. Soittola, Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science (Springer, New York, 1978), pp. x+171CrossRefGoogle Scholar
  32. [vdL04]
    P. van der Laan, Operads: Hopf Algebras and Coloured Koszul Duality, Ph.D. thesis, Universiteit Utrecht, 2004Google Scholar
  33. [Yau16]
    D. Yau, Colored Operads. Graduate Studies in Mathematics (American Mathematical Society, Providence, 2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Samuele Giraudo
    • 1
  1. 1.University of Paris-EstMarne-la-ValleeFrance

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