Advertisement

The Ramanujan Journal

, Volume 31, Issue 1–2, pp 33–51 | Cite as

A multivariate “inv” hook formula for forests

  • Florent Hivert
  • Victor Reiner
Article

Abstract

Björner and Wachs provided two q-generalizations of Knuth’s hook formula counting linear extensions of forests: one involving the major index statistic, and one involving the inversion number statistic. We prove a multivariate generalization of their inversion number result, motivated by specializations related to the modular invariant theory of finite general linear groups.

Keywords

Hook formula Forests Moulds Binary search Free quasisymmetric functions Loday–Ronco algebra 

Mathematics Subject Classification

05A15 05A10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Björner, A., Wachs, M.L.: q-hook length formulas for forests. J. Comb. Theory, Ser. A 52(2), 165–187 (1989) MATHCrossRefGoogle Scholar
  2. 2.
    Boussicault, A., Feray, V., Lascoux, A., Reiner, V.: Linear extension sums as valuations of cones. J. Algeb. Comb., to appear Google Scholar
  3. 3.
    Chapoton, F., Hivert, F., Novelli, J.-C., Thibon, J.-Y.: An operational calculus for the mould operad. Int. Math. Res. Not. IMRN 9, rnn018 (2008). 22 pp MathSciNetGoogle Scholar
  4. 4.
    Hivert, F., Novelli, J.-C., Thibon, J.-Y.: The algebra of binary search trees. Theor. Comput. Sci. 339(1), 129–165 (2005) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hivert, F., Novelli, J.-C., Thibon, J.-Y.: Trees, functional equations, and combinatorial Hopf algebras. Eur. J. Comb. 29(7), 1682–1695 (2008) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Knuth, D.E.: Sorting and searching. In: The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1973) Google Scholar
  7. 7.
    Loday, J.-L., Ronco, M.O.: Hopf algebra of the planar binary trees. Adv. Math. 139(2), 293–309 (1998) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177, 967–982 (1995) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Reiner, V., Stanton, D.: (q,t)-analogues and \(\mathit{GL}_{n}(\mathbb{F}_{q})\). J. Algebr. Comb. 31, 411–454 (2010) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (1997) MATHCrossRefGoogle Scholar
  11. 11.
    Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Laboratoire de Recherche en Informatique (UMR CNRS 8623)Orsay CedexFrance
  2. 2.LITISUniversité de RouenSaint Étienne du RouvrayFrance
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations