Advertisement

Weighted Staircase Tableaux, Asymmetric Exclusion Process, and Eulerian Type Recurrences

  • Paweł Hitczenko
  • Svante Janson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

We consider a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey–Wilson polynomials; however, their purely combinatorial properties have gained considerable interest in the past few years.

We will be interested in a general model of staircase tableaux in which symbols that appear in staircase tableaux may have arbitrary positive weights. Under this general model we derive a number of results concerning the limiting laws for the number of appearances of symbols in a random staircase tableaux.

One advantage of our generality is that we may let the weights approach extreme values of zero or infinity, which covers further special cases appearing earlier in the literature.

One of the main tools we use are generating functions of the parameters of interests. This leads us to a two–parameter family of polynomials. Specific values of the parameters cover a number of special cases analyzed earlier in the literature including the classical Eulerian polynomials.

Keywords

Staircase tableau Eulerian polynomial Asymmetric Exclusion Process 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aval, J.-C., Boussicault, A., Nadeau, P.: Tree-like tableaux. In: 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011). Discrete Math. Theor. Comput. Sci. Proc., AO, pp. 63–74 (2011)Google Scholar
  2. 2.
    Carlitz, L., Scoville, R.: Generalized Eulerian numbers: combinatorial applications. J. Reine Angew. Math. 265, 110–137 (1974)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Corteel, S., Dasse-Hartaut, S.: Statistics on staircase tableaux, Eulerian and Mahonian statistics. In: 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, pp. 245–255 (2011)Google Scholar
  4. 4.
    Corteel, S., Hitczenko, P.: Expected values of statistics on permutation tableaux. In: 2007 Conference on Analysis of Algorithms, AofA 2007, Discrete Math. Theor. Comput. Sci. Proc., AH, pp. 325–339 (2007)Google Scholar
  5. 5.
    Corteel, S., Stanley, R., Stanton, D., Williams, L.: Formulae for Askey–Wilson moments and enumeration of staircase tableaux. Trans. Amer. Math. Soc. 364(11), 6009–6037 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Corteel, S., Williams, L.K.: A Markov chain on permutations which projects to the PASEP. Int. Math. Res. Notes, Article 17:rnm055, 27pp (2007)Google Scholar
  7. 7.
    Corteel, S., Williams, L.K.: Tableaux combinatorics for the asymmetric exclusion process. Adv. Appl. Math. 39, 293–310 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Corteel, S., Williams, L.K.: Staircase tableaux, the asymmetric exclusion process, and Askey–Wilson polynomials. Proc. Natl. Acad. Sci. 107(15), 6726–6730 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Corteel, S., Williams, L.K.: Tableaux combinatorics for the asymmetric exclusion process and Askey–Wilson polynomials. Duke Math. J. 159, 385–415 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Dasse-Hartaut, S., Hitczenko, P.: Greek letters in random staircase tableaux. Random Struct. Algorithms 42, 73–96 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Derrida, B., Evans, M.R., Hakim, V., Pasquier, V.: Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26(7), 1493–1517 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Franssens, G.R.: On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles. J. Integer Seq. 9(4):Article 06.4.1, 34 (2006)Google Scholar
  13. 13.
    Frobenius, G.: Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, pp. 809–847 (1910)Google Scholar
  14. 14.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  15. 15.
    Hitczenko, P., Janson, S.: Asymptotic normality of statistics on permutation tableaux. Contemporary Math. 520, 83–104 (2010)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Hitczenko, P., Janson, S.: Weighted random staircase tableaux. To appear in Combin. Probab. Comput., arxiv.org/abs/1212.5498Google Scholar
  17. 17.
    Janson, S.: Euler–Frobenius numbers and rounding. arxiv.org/abs/1305.3512Google Scholar
  18. 18.
    Liu, L.L., Wang, Y.: A unified approach to polynomial sequences with only real zeros. Adv. Appl. Math. 38(4), 542–560 (2007)CrossRefzbMATHGoogle Scholar
  19. 19.
    MacMahon, P.A.: The divisors of numbers. Proc. London Math. Soc. Ser. 2 19(1), 305–340 (1920)zbMATHGoogle Scholar
  20. 20.
    Nadeau, P.: The structure of alternative tableaux. J. Combin. Theory Ser. A 118(5), 1638–1660 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/
  22. 22.
    The On-Line Encyclopedia of Integer Sequences, http://oeis.org
  23. 23.
    Petrov, V.V.: Sums of Independent Random Variables. Springer, Berlin (1975)CrossRefGoogle Scholar
  24. 24.
    Stanley, R.P.: Enumerative Combinatorics, vol. I. Cambridge Univ. Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, Y., Yeh, Y.-N.: Polynomials with real zeros and Pólya frequency sequences. J. Combin. Theory Ser. A 109(1), 63–74 (2005)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Paweł Hitczenko
    • 1
  • Svante Janson
    • 2
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

Personalised recommendations