Skip to main content Accessibility help

Weighted Random Staircase Tableaux



This paper concerns 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.

In this paper we further study combinatorial properties of staircase tableaux. We consider a general model of random staircase tableaux in which symbols (Greek letters) that appear in staircase tableaux may have arbitrary positive weights. (We consider only the case with the parameters u = q = 1.) Under this general model we derive a number of results. Some of our results concern the limiting laws for the number of appearances of symbols in a random staircase tableaux. They generalize and subsume earlier results that were obtained for specific values of the weights.

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. Furthermore, our generality allows us to analyse the structure of random staircase tableaux, and we obtain several results in this direction.

One of the tools we use is the generating functions of the parameters of interest. This leads us to a two-parameter family of polynomials, generalizing the classical Eulerian polynomials.

We also briefly discuss the relation of staircase tableaux to the asymmetric exclusion process, to other recently introduced types of tableaux, and to an urn model studied by a number of researchers, including Philippe Flajolet.



Hide All
[1]Aval, J.-C., Boussicault, A. and Nadeau, P. (2011) Tree-like tableaux. In 23rd International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2011, DMTCS proc. AO 63–74.
[2]Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation. Oxford University Press.
[3]Barbour, A. and Janson, S. (2009) A functional combinatorial central limit theorem. Electron. J. Probab. 14, #81, 23522370.
[4]Bernstein, S. (1940) Nouvelles applications des grandeurs aléatoires presqu'indépendantes (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 4 137150.
[5]Bernstein, S. (1940) Sur un problème du schéma des urnes à composition variable. CR (Doklady) Acad. Sci. URSS (NS) 28 57.
[6]Brenti, F. (1994) q-Eulerian polynomials arising from Coxeter groups. European J. Combin. 15 417441.
[7]Carlitz, L. (1959) Eulerian numbers and polynomials. Mathematics Magazine 32 247260.
[8]Carlitz, L., Kurtz, D. C., Scoville, R. and Stackelberg, O. P. (1972) Asymptotic properties of Eulerian numbers. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 23 4754.
[9]Carlitz, L. and Scoville, R. (1974) Generalized Eulerian numbers: combinatorial applications. J. Reine Angew. Math. 265 110137.
[10]Chow, C.-O. and Gessel, I. M. (2007) On the descent numbers and major indices for the hyperoctahedral group. Adv. Appl. Math. 38 275301.
[11]Corteel, S. and Dasse-Hartaut, S. (2011) Statistics on staircase tableaux, Eulerian and Mahonian statistics. In 23rd International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2011, DMTCS proc. AO 245–255.
[12]Corteel, S. and Hitczenko, P. (2007) Expected values of statistics on permutation tableaux. In 2007 Conference on Analysis of Algorithms: AofA 07, DMTCS proc. AH 325–339.
[13]Corteel, S. and Nadeau, P. (2009) Bijections for permutation tableaux. European J. Combin. 30 295310.
[14]Corteel, S., Stanley, R., Stanton, D. and Williams, L. (2012) Formulae for Askey–Wilson moments and enumeration of staircase tableaux. Trans. Amer. Math. Soc. 364 60096037.
[15]Corteel, S. and Williams, L. K. (2007) A Markov chain on permutations which projects to the PASEP. Int. Math. Res. Notes, #17.
[16]Corteel, S. and Williams, L. K. (2007) Tableaux combinatorics for the asymmetric exclusion process. Adv. Appl. Math. 39 293310.
[17]Corteel, S. and Williams, L. K. (2010) Staircase tableaux, the asymmetric exclusion process, and Askey–Wilson polynomials. Proc. Natl Acad. Sci. 107 67266730.
[18]Corteel, S. and Williams, L. K. (2011) Tableaux combinatorics for the asymmetric exclusion process and Askey–Wilson polynomials. Duke Math. J., 159: 385415.
[19]Dasse-Hartaut, S. and Hitczenko, P. (2013) Greek letters in random staircase tableaux. Random Struct. Alg. 42 7396.
[20]Derrida, B., Evans, M. R., Hakim, V. and Pasquier, V. (1993) Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26 14931517.
[21]Euler, L. (1736) Methodus universalis series summandi ulterius promota. In Commentarii Academiae Acientiarum Imperialis Petropolitanae 8, St. Petersburg (1741), pp. 147–158.
[22]Euler, L. (1755) Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum. Vol I. St. Petersburg.
[23]Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popul. Biol. 3 87112.
[24]Féray, V. (2013) Asymptotic behavior of some statistics in Ewens random permutations. Electron. J. Probab. 18 (76), 132.
[25]Flajolet, P., Dumas, P. and Puyhaubert, V. (2006) Some exactly solvable models of urn process theory. In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS proc. AG 59–118.
[26]Franssens, G. R. (2006) On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles. J. Integer Seq. 9 #06.4.1.
[27]Freedman, D. A. (1965) Bernard Friedman's urn. Ann. Math. Statist. 36 956970.
[28]Friedman, B. (1949) A simple urn model. Comm. Pure Appl. Math. 2 5970.
[29]Frobenius, G. (1910) Über die Bernoullischen Zahlen und die Eulerschen Polynome. In Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, pp. 809847.
[30]Gawronski, W. and Neuschel, T. (2013) Euler–Frobenius numbers. Integral Transforms and Special Functions 24 817830.
[31]Graham, R. L., Knuth, D. E. and Patashnik, O. (1994) Concrete Mathematics, second edition, Addison-Wesley.
[32]Gut, A. (2013) Probability: A Graduate Course, second edition, Springer.
[33]Hitczenko, P. and Janson, S. (2010) Asymptotic normality of statistics on permutation tableaux. Contemporary Math. 520 83104.
[34]Janson, S. (2004) Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Proc. Appl. 110 177245.
[35]Janson, S. (2013) Euler–Frobenius polynomials and rounding. Online Journal of Analytic Combinatorics, [S.l.] 8.
[36]Liu, L. L. and Wang, Y. (2007) A unified approach to polynomial sequences with only real zeros. Adv. Appl. Math. 38 542560.
[37]MacMahon, P. A. (1920) The divisors of numbers. Proc. London Math. Soc. Ser. 2 19 305340.
[38]Meinardus, G. and Merz, G. (1974) Zur periodischen Spline-Interpolation. In Spline-Funktionen: Oberwolfach 1973, Bibliographisches Institut, pp. 177195.
[39]Nadeau, P. (2011) The structure of alternative tableaux. J. Combin. Theory Ser. A 118 16381660.
[40]NIST Digital Library of Mathematical Functions.
[41]The On-Line Encyclopedia of Integer Sequences.
[42]Petrov, V. V. (1975) Sums of Independent Random Variables, Springer.
[43]Reimer, M. (1982) Extremal spline bases. J. Approx. Theory 36 9198.
[44]Reimer, M. (1985) The main roots of the Euler–Frobenius polynomials. J. Approx. Theory 45 358362.
[45]Schmidt, F. and Simion, R. (1997) Some geometric probability problems involving the Eulerian numbers. Electron. J. Combin. 4 R18.
[46]Siepmann, D. (1988) Cardinal interpolation by polynomial splines: interpolation of data with exponential growth. J. Approx. Theory 53 167183.
[47]Stanley, R. P. (1997) Enumerative Combinatorics, Vol. I, Cambridge University Press.
[48]Steingrímsson, E. and Williams, L. K. (2007) Permutation tableaux and permutation patterns. J. Combin. Theory Ser. A 114 211234.
[49]ter Morsche, H. (1974) On the existence and convergence of interpolating periodic spline functions of arbitrary degree. In Spline-Funktionen: Oberwolfach 1973, Bibliographisches Institut, pp. 197214.
[50]Wang, Y. and Yeh, Y.-N. (2005) Polynomials with real zeros and Pólya frequency sequences. J. Combin. Theory Ser. A 109 6374.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed