Advertisement

Journal of Algebraic Combinatorics

, Volume 36, Issue 3, pp 333–354 | Cite as

Double Catalan monoids

  • Volodymyr Mazorchuk
  • Benjamin Steinberg
Article

Abstract

In this paper, we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results of Barnabei et al. on admissible pairs of Dyck paths. We compute a presentation and determine the minimal dimension of an effective representation for the double Catalan monoid. We also determine the minimal dimension of an effective representation for the 0-Hecke monoid.

Keywords

Catalan monoid Presentation Pattern avoiding permutation Effective representation 

References

  1. 1.
    Abramenko, P., Brown, K.: Buildings. Theory and Applications. Graduate Texts in Mathematics, vol. 248. Springer, New York (2008) MATHGoogle Scholar
  2. 2.
    Barnabei, M., Bonetti, F., Silimbani, M.: Restricted involutions and Motzkin paths. Adv. Appl. Math. 47(1), 102–115 (2011) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Barnabei, M., Bonetti, F., Silimbani, M.: 1234-avoiding permutations and Dyck paths. Preprint. arXiv:1102.1541
  4. 4.
    Billey, S., Jockusch, W., Stanley, R.: Some combinatorial properties of Schubert polynomials. J. Algebr. Comb. 2(4), 345–374 (1993) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005) MATHGoogle Scholar
  6. 6.
    Bóna, M.: Combinatorics of Permutations. Discrete Mathematics and Its Applications. Chapman & Hall/CRC, Boca Raton (2004) MATHCrossRefGoogle Scholar
  7. 7.
    Carter, R.: Representation theory of the 0-Hecke algebra. J. Algebra 104(1), 89–103 (1986) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Denton, T.: Zero-Hecke monoids and pattern avoidance. Available at: http://oz.plymouth.edu/~dcernst/SpecialSession/Denton.pdf
  9. 9.
    Fayers, M.: 0-Hecke algebras of finite Coxeter groups. J. Pure Appl. Algebra 199(1–3), 27–41 (2005) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ganyushkin, O., Mazorchuk, V.: Factor powers of finite symmetric groups. Mat. Zametki 58(2), 176–188 (1995). Translation in Math. Notes 58(1–2), 794–802 (1996) MathSciNetGoogle Scholar
  11. 11.
    Ganyushkin, O., Mazorchuk, V.: Classical Finite Transformation Semigroups. An Introduction. Algebra and Applications, vol. 9. Springer, London (2009) MATHCrossRefGoogle Scholar
  12. 12.
    Ganyushkin, O., Mazorchuk, V.: On Kiselman quotients of 0-Hecke monoids. Int. Electron. J. Algebra 10, 174–191 (2011) MathSciNetGoogle Scholar
  13. 13.
    Gessel, I.: Symmetric functions and P-recursiveness. J. Comb. Theory, Ser. A 53(2), 257–285 (1990) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hivert, F., Novelli, J.-C., Thibon, J.-Y.: Yang–Baxter bases of 0-Hecke algebras and representation theory of 0-Ariki–Koike–Shoji algebras. Adv. Math. 205(2), 504–548 (2006) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hivert, F., Schilling, A., Thiéry, N.: Hecke group algebras as degenerate affine Hecke algebras. In: 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008). Discrete Math. Theor. Comput. Sci. Proc., pp. 611–623. AJ Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2008) Google Scholar
  16. 16.
    Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165–184 (1979) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kim, K.H., Roush, F.: Linear representations of semigroups of Boolean matrices. Proc. Am. Math. Soc. 63(2), 203–207 (1977) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kreweras, G.: Sur les éventails de segments. Cah. BURO 15, 3–41 (1970) Google Scholar
  19. 19.
    Margolis, S., Steinberg, B.: Quivers of monoids with basic algebras. Preprint. arXiv:1101.0416
  20. 20.
    Mathas, A.: Iwahori–Hecke algebras and Schur algebras of the symmetric group. University Lecture Series, vol. 15. American Mathematical Society, Providence (1999) MATHGoogle Scholar
  21. 21.
    McNamara, P.: EL-labelings, supersolvability and 0-Hecke algebra actions on posets. J. Comb. Theory, Ser. A 101(1), 69–89 (2003) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Norton, P.: 0-Hecke algebras. J. Aust. Math. Soc. A 27(3), 337–357 (1979) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Novelli, J.-C., Thibon, J.-Y.: Noncommutative symmetric functions and Lagrange inversion. Adv. Appl. Math. 40(1), 8–35 (2008) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Richardson, R., Springer, T.: The Bruhat order on symmetric varieties. Geom. Dedic. 35(1–3), 389–436 (1990) MathSciNetMATHGoogle Scholar
  25. 25.
    Sloane, N.: The Online Encyclopedia of Integer Sequences. http://oeis.org/
  26. 26.
    Solomon, A.: Catalan monoids, monoids of local endomorphisms, and their presentations. Semigroup Forum 53(3), 351–368 (1996) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Stembridge, J.: A short derivation of the Möbius function for the Bruhat order. J. Algebr. Comb. 25(2), 141–148 (2007) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Straubing, H., Thérien, D.: Partially ordered finite monoids and a theorem of I. Simon. J. Algebra 119(2), 393–399 (1988) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Tenner, B.: Reduced decompositions and permutation patterns. J. Algebr. Comb. 24(3), 263–284 (2006) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  3. 3.Department of MathematicsCity College of New YorkNew YorkUSA

Personalised recommendations