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Weighted Random Staircase Tableaux

Published online by Cambridge University Press:  15 July 2014

PAWEŁ HITCZENKO  and
SVANTE JANSON
PAWEŁ HITCZENKO
Affiliation:
Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA (e-mail: phitczenko@math.drexel.edu, http://www.math.drexel.edu/~phitczen/)
SVANTE JANSON
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden (e-mail: svante.janson@math.uu.se, http://www2.math.uu.se/~svante/)
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Abstract

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.

Keywords

Primary 60C05Secondary 05A1505E9960F05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 6: Honouring the Memory of Philippe Flajolet - Part 2 , November 2014 , pp. 1114 - 1147
Copyright
Copyright © Cambridge University Press 2014 

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References

[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.Google Scholar
[2]Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation. Oxford University Press.Google Scholar
[3]Barbour, A. and Janson, S. (2009) A functional combinatorial central limit theorem. Electron. J. Probab. 14, #81, 2352–2370.Google Scholar
[4]Bernstein, S. (1940) Nouvelles applications des grandeurs aléatoires presqu'indépendantes (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 4 137–150.Google Scholar
[5]Bernstein, S. (1940) Sur un problème du schéma des urnes à composition variable. CR (Doklady) Acad. Sci. URSS (NS) 28 5–7.Google Scholar
[6]Brenti, F. (1994) q-Eulerian polynomials arising from Coxeter groups. European J. Combin. 15 417–441.Google Scholar
[7]Carlitz, L. (1959) Eulerian numbers and polynomials. Mathematics Magazine 32 247–260.CrossRefGoogle Scholar
[8]Carlitz, L., Kurtz, D. C., Scoville, R. and Stackelberg, O. P. (1972) Asymptotic properties of Eulerian numbers. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 23 47–54.CrossRefGoogle Scholar
[9]Carlitz, L. and Scoville, R. (1974) Generalized Eulerian numbers: combinatorial applications. J. Reine Angew. Math. 265 110–137.Google Scholar
[10]Chow, C.-O. and Gessel, I. M. (2007) On the descent numbers and major indices for the hyperoctahedral group. Adv. Appl. Math. 38 275–301.Google Scholar
[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.CrossRefGoogle Scholar
[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.Google Scholar
[13]Corteel, S. and Nadeau, P. (2009) Bijections for permutation tableaux. European J. Combin. 30 295–310.Google Scholar
[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 6009–6037.Google Scholar
[15]Corteel, S. and Williams, L. K. (2007) A Markov chain on permutations which projects to the PASEP. Int. Math. Res. Notes, #17.Google Scholar
[16]Corteel, S. and Williams, L. K. (2007) Tableaux combinatorics for the asymmetric exclusion process. Adv. Appl. Math. 39 293–310.Google Scholar
[17]Corteel, S. and Williams, L. K. (2010) Staircase tableaux, the asymmetric exclusion process, and Askey–Wilson polynomials. Proc. Natl Acad. Sci. 107 6726–6730.Google Scholar
[18]Corteel, S. and Williams, L. K. (2011) Tableaux combinatorics for the asymmetric exclusion process and Askey–Wilson polynomials. Duke Math. J., 159: 385–415.Google Scholar
[19]Dasse-Hartaut, S. and Hitczenko, P. (2013) Greek letters in random staircase tableaux. Random Struct. Alg. 42 73–96.Google Scholar
[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 1493–1517.CrossRefGoogle Scholar
[21]Euler, L. (1736) Methodus universalis series summandi ulterius promota. In Commentarii Academiae Acientiarum Imperialis Petropolitanae 8, St. Petersburg (1741), pp. 147–158. http://www.math.dartmouth.edu/~euler/pages/E055.htmlGoogle Scholar
[22]Euler, L. (1755) Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum. Vol I. St. Petersburg. http://www.math.dartmouth.edu/~euler/pages/E212.htmlGoogle Scholar
[23]Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popul. Biol. 3 87–112.CrossRefGoogle ScholarPubMed
[24]Féray, V. (2013) Asymptotic behavior of some statistics in Ewens random permutations. Electron. J. Probab. 18 (76), 1–32.CrossRefGoogle Scholar
[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.Google Scholar
[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.Google Scholar
[27]Freedman, D. A. (1965) Bernard Friedman's urn. Ann. Math. Statist. 36 956–970.Google Scholar
[28]Friedman, B. (1949) A simple urn model. Comm. Pure Appl. Math. 2 59–70.Google Scholar
[29]Frobenius, G. (1910) Über die Bernoullischen Zahlen und die Eulerschen Polynome. In Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, pp. 809–847.Google Scholar
[30]Gawronski, W. and Neuschel, T. (2013) Euler–Frobenius numbers. Integral Transforms and Special Functions 24 817–830.Google Scholar
[31]Graham, R. L., Knuth, D. E. and Patashnik, O. (1994) Concrete Mathematics, second edition, Addison-Wesley.Google Scholar
[32]Gut, A. (2013) Probability: A Graduate Course, second edition, Springer.Google Scholar
[33]Hitczenko, P. and Janson, S. (2010) Asymptotic normality of statistics on permutation tableaux. Contemporary Math. 520 83–104.Google Scholar
[34]Janson, S. (2004) Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Proc. Appl. 110 177–245.Google Scholar
[35]Janson, S. (2013) Euler–Frobenius polynomials and rounding. Online Journal of Analytic Combinatorics, [S.l.] 8.Google Scholar
[36]Liu, L. L. and Wang, Y. (2007) A unified approach to polynomial sequences with only real zeros. Adv. Appl. Math. 38 542–560.Google Scholar
[37]MacMahon, P. A. (1920) The divisors of numbers. Proc. London Math. Soc. Ser. 2 19 305–340.Google Scholar
[38]Meinardus, G. and Merz, G. (1974) Zur periodischen Spline-Interpolation. In Spline-Funktionen: Oberwolfach 1973, Bibliographisches Institut, pp. 177–195.Google Scholar
[39]Nadeau, P. (2011) The structure of alternative tableaux. J. Combin. Theory Ser. A 118 1638–1660.Google Scholar
[40]NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/Google Scholar
[41]The On-Line Encyclopedia of Integer Sequences. http://oeis.orgGoogle Scholar
[42]Petrov, V. V. (1975) Sums of Independent Random Variables, Springer.Google Scholar
[43]Reimer, M. (1982) Extremal spline bases. J. Approx. Theory 36 91–98.Google Scholar
[44]Reimer, M. (1985) The main roots of the Euler–Frobenius polynomials. J. Approx. Theory 45 358–362.CrossRefGoogle Scholar
[45]Schmidt, F. and Simion, R. (1997) Some geometric probability problems involving the Eulerian numbers. Electron. J. Combin. 4 R18.Google Scholar
[46]Siepmann, D. (1988) Cardinal interpolation by polynomial splines: interpolation of data with exponential growth. J. Approx. Theory 53 167–183.Google Scholar
[47]Stanley, R. P. (1997) Enumerative Combinatorics, Vol. I, Cambridge University Press.Google Scholar
[48]Steingrímsson, E. and Williams, L. K. (2007) Permutation tableaux and permutation patterns. J. Combin. Theory Ser. A 114 211–234.Google Scholar
[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. 197–214.Google Scholar
[50]Wang, Y. and Yeh, Y.-N. (2005) Polynomials with real zeros and Pólya frequency sequences. J. Combin. Theory Ser. A 109 63–74.Google Scholar