Advertisement

Journal of Algebraic Combinatorics

, Volume 45, Issue 3, pp 857–886 | Cite as

Semi-pointed partition posets and species

  • Bérénice Delcroix-Oger
Article

Abstract

We define semi-pointed partition posets, which are a generalization of partition posets, and show that they are Cohen–Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Faà di Bruno Hopf algebra.

Keywords

Poset Incidence Hopf algebra Möbius number Partitions Species Operad 

References

  1. 1.
    Bergeron, F., Labelle, G., Leroux, P.: Combinatorial Species and Tree-like Structures, vol. 67 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge. Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota (1998)Google Scholar
  2. 2.
    Boardman, J.M., Vogt, R.M.: Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347. Springer, Berlin (1973)CrossRefzbMATHGoogle Scholar
  3. 3.
    Björner, A., Wachs, M.: On lexicographically shellable posets. Trans. Am. Math. Soc. 277(1), 323–341 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chapoton, F., Livernet, M.: Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 8, 395–408 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chapoton, F., Vallette, B.: Pointed and multi-pointed partitions of type \(A\) and \(B\). J. Algebr. Comb. 23(4), 295–316 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Constantine, G.M., Savits, T.H.: A multivariate Faa di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Delcroix-Oger, B.: Hyperarbres et Partitions semi-pointées: Aspects combinatoires, algébriques et homologiques. PhD thesis, Université Lyon 1 (2014)Google Scholar
  8. 8.
    Delcroix-Oger, B.: Semi-pointed partition posets. In: Proceedings of the FPSAC 15 Conference, DMTCS (2015)Google Scholar
  9. 9.
    Edelman, P.H.: Zeta polynomials and the Möbius function. Eur. J. Comb. 1(4), 335–340 (1980)CrossRefzbMATHGoogle Scholar
  10. 10.
    Elmendorf, A.D., Mandell, M.A.: Rings, modules, and algebras in infinite loop space theory. Adv. Math. 205(1), 163–228 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fresse, B.: Koszul duality of operads and homology of partition posets. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic \(K\)-Theory, vol. 346 of Contemp. Math., pp. 115–215. Am. Math. Soc., Providence, RI (2004)Google Scholar
  12. 12.
    Getzler, E.: Operads and moduli spaces of genus \(0\) Riemann surfaces. In: The Moduli Space of Curves (Texel Island, 1994), vol. 129 of Prog. Math., pp. 199–230. Birkhäuser Boston, Boston, MA (1995)Google Scholar
  13. 13.
    Hanlon, P.: The fixed-point partition lattices. Pac. J. Math. 96(2), 319–341 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Joyal, A.: Foncteurs analytiques et espèces de structures. In: Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985), vol. 1234 of Lecture Notes in Math., pp. 126–159. Springer, Berlin (1986)Google Scholar
  15. 15.
    Loday, J.-L., Vallette, B.: Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346. Springer, Heidelberg (2012)Google Scholar
  16. 16.
    May, J.P.: The Geometry of Iterated Loop Spaces. Lectures Notes in Mathematics, Vol. 271. Springer, Berlin (1972)Google Scholar
  17. 17.
    Oger, B.: Action of the symmetric groups on the homology of the hypertree posets. J. Algebr. Comb. 38(4), 915–945 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Oger, B.: Incidence Hopf algebra of the hypertree posets. Sém. Lothar. Combin. 72, 22 (2014)Google Scholar
  19. 19.
    Robinson, A., Whitehouse, S.: The tree representation of \(\sigma _n+1\). J. Pure Appl. Algebra 111(1–3), 245–253 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schmitt, W.R.: Antipodes and incidence coalgebras. J. Comb. Theory Ser. A 46(2), 264–290 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schmitt, W.R.: Incidence Hopf algebras. J. Pure Appl. Algebra 96(3), 299–330 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Stanley, R.P.: Combinatorial reciprocity theorems. Adv. Math. 14, 194–253 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Stanley, R.P.: Some aspects of groups acting on finite posets. J. Comb. Theory Ser. A 32(2), 132–161 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Stanley, R.P.: Enumerative Combinatorics, vol. 2, vol. 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press (2001)Google Scholar
  25. 25.
    Stanley, R.P.: Enumerative Combinatorics, vol. 1, vol. 49 of Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2012)Google Scholar
  26. 26.
    Vallette, B.: Homology of generalized partition posets. J. Pure Appl. Algebra 208(2), 699–725 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Van der Laan, P.: Operads: Hopf Algebras and Coloured Koszul Duality. PhD thesis, Utrecht University (2004)Google Scholar
  28. 28.
    Wachs, M.L.: Poset topology: tools and applications. In: Geometric Combinatorics, vol. 13 of IAS/Park City Math. Ser., pp. 497–615. Am. Math. Soc., Providence, RI (2007)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institut Camille Jordan, UMR 5208Université Claude Bernard Lyon 1Villeurbanne CedexFrance

Personalised recommendations