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Classifying Discrete Structures by Their Stabilizers

  • Gilbert LabelleEmail author
Conference paper
  • 54 Downloads
Part of the Communications in Computer and Information Science book series (CCIS, volume 1125)

Abstract

Combinatorial power series are formal power series of the form \(\sum c_{n,H}X^n/H \) where, for each n, H runs through subgroups of the symmetric group \(S_n\) and the coefficients \(c_{n,H}\) are complex numbers (or ordinary power series involving some “weight variables”). Such series conveniently encode species of combinatorial (possibly weighted) structures according to their stabilizers (up to conjugacy). We give general lines for expressing these kinds of series – as well as the main operations \((+,\cdot ,\times ,\circ ,d/dX)\) between them – by making use of the GroupTheory package and give suggestions for possible extensions of that package and some other specific procedures such as collect, expand, series, etc. An analysis of multivariable combinatorial power series is also presented.

Keywords

Discrete structures Stabilizers Combinatorial operations 

References

  1. 1.
    Bergeron, F., Labelle, G., Leroux, P.: Combinatorial Species and Tree-like Structures. Ency. of Mathematics and Its Applications, vol. 67. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  2. 2.
    Joyal, A.: Une théorie combinatoire des séries formelles. Adv. Math. 42, 1–82 (1981)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Labelle, G.: New combinatorial computational methods arising from pseudo singletons. In: Discrete Mathematics and Theoretical Computer Science, pp. 247–258 (2008)Google Scholar
  4. 4.
    Labelle, G.: Binomial species and combinatorial exponentiation. J. Électronique du Séminaire Lotharingien de Combinatoire 78, B78a (2018)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Yeh, Y.-N.: The calculus of virtual species and K-species. In: Labelle, G., Leroux, P. (eds.) Combinatoire énumérative. LNM, vol. 1234, pp. 351–369. Springer, Heidelberg (1986).  https://doi.org/10.1007/BFb0072525. ISBN 978-3-540-47402-9CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.LaCIM, Université du Québec à MontréalMontréalCanada

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