Formulae for Askey-Wilson moments and enumeration of staircase tableaux

Corteel, S.; Stanley, R.; Stanton, D.; Williams, L.

doi:10.1090/S0002-9947-2012-05588-7

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

Formulae for Askey-Wilson moments and enumeration of staircase tableaux

Authors: S. Corteel, R. Stanley, D. Stanton and L. Williams
Journal: Trans. Amer. Math. Soc. 364 (2012), 6009-6037
MSC (2010): Primary 05A15; Secondary 33C45, 82B23
Posted: May 2, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We explain how the moments of the (weight function of the) Askey-Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors. This gives us a direct combinatorial formula for these moments, which is related to, but more elegant than the formula given in their earlier paper. Then we use techniques developed by Ismail and the third author to give explicit formulae for these moments and for the enumeration of staircase tableaux. Finally we study the enumeration of staircase tableaux at various specializations of the parameterizations; for example, we obtain the Catalan numbers, Fibonacci numbers, Eulerian numbers, the number of permutations, and the number of matchings.

1. Richard Askey, Beta integrals and the associated orthogonal polynomials, Number theory, Madras 1987, Lecture Notes in Math., vol. 1395, Springer, Berlin, 1989, pp. 84–121. MR 1019328 (90k:33001), http://dx.doi.org/10.1007/BFb0086401
2. Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216 (87a:05023)
3. J. C. Aval, A. Boussicault and S. Dasse-Hartaut, The tree structure in staircase tableaux, Gascom 2012. arxiv:1109.4907.
4. R. Brak and J. W. Essam, Asymmetric exclusion model and weighted lattice paths, J. Phys. A 37 (2004), no. 14, 4183–4217. MR 2066074 (2005f:82040), http://dx.doi.org/10.1088/0305-4470/37/14/002
5. Alexander Burstein, On some properties of permutation tableaux, Ann. Comb. 11 (2007), no. 3-4, 355–368. MR 2376110 (2008m:05003), http://dx.doi.org/10.1007/s00026-007-0323-0
6. T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York, 1978. Mathematics and its Applications, Vol. 13. MR 0481884 (58 #1979)
7. Sylvie Corteel, Crossings and alignments of permutations, Adv. in Appl. Math. 38 (2007), no. 2, 149–163. MR 2290808 (2007j:05003), http://dx.doi.org/10.1016/j.aam.2006.01.006
8. Sylvie Corteel, Matthieu Josuat-Vergès, and Lauren K. Williams, The matrix ansatz, orthogonal polynomials, and permutations, Adv. in Appl. Math. 46 (2011), no. 1-4, 209–225. MR 2794022 (2012i:05013), http://dx.doi.org/10.1016/j.aam.2010.04.009
9. Sylvie Corteel and Philippe Nadeau, Bijections for permutation tableaux, European J. Combin. 30 (2009), no. 1, 295–310. MR 2460235 (2009k:05191), http://dx.doi.org/10.1016/j.ejc.2007.12.007
10. Sylvie Corteel and Lauren K. Williams, Tableaux combinatorics for the asymmetric exclusion process, Adv. in Appl. Math. 39 (2007), no. 3, 293–310. MR 2352041 (2008g:05220), http://dx.doi.org/10.1016/j.aam.2006.08.002
11. S. Corteel and L.K. Williams, A Markov chain on permutations which projects to the asymmetric exclusion process, Int. Math. Res. Not. (2007), no. 17, Art. ID rnm055, 27 pp.
12. Sylvie Corteel and Lauren K. Williams, Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials, Proc. Natl. Acad. Sci. USA 107 (2010), no. 15, 6726–6730. MR 2630104 (2011k:05268), http://dx.doi.org/10.1073/pnas.0909915107
13. Sylvie Corteel and Lauren K. Williams, Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials, Duke Math. J. 159 (2011), no. 3, 385–415. MR 2831874 (2012h:05359), http://dx.doi.org/10.1215/00127094-1433385
14. Sylvie Corteel and Sandrine Dasse-Hartaut, Statistics on staircase tableaux, Eulerian and Mahonian statistics, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2011, pp. 245–255 (English, with English and French summaries). MR 2820714
15. B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26 (1993), no. 7, 1493–1517. MR 1219679 (94g:60179)
16. Enrica Duchi and Gilles Schaeffer, A combinatorial approach to jumping particles, J. Combin. Theory Ser. A 110 (2005), no. 1, 1–29. MR 2128962 (2006c:60130), http://dx.doi.org/10.1016/j.jcta.2004.09.006
17. Philippe Flajolet, On congruences and continued fractions for some classical combinatorial quantities, Discrete Math. 41 (1982), no. 2, 145–153. MR 676874 (84f:05005), http://dx.doi.org/10.1016/0012-365X(82)90201-1
18. P. Flajolet, J. Françon, and J. Vuillemin, Sequence of operations analysis for dynamic data structures, J. Algorithms 1 (1980), no. 2, 111–141. MR 604861 (82m:68039), http://dx.doi.org/10.1016/0196-6774(80)90020-6
19. George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719 (2006d:33028)
20. Mourad E. H. Ismail and Dennis Stanton, 𝑞-Taylor theorems, polynomial expansions, and interpolation of entire functions, J. Approx. Theory 123 (2003), no. 1, 125–146. MR 1985020 (2004g:30040), http://dx.doi.org/10.1016/S0021-9045(03)00076-5
21. Matthieu Josuat-Vergès, Combinatorics of the three-parameter PASEP partition function, Electron. J. Combin. 18 (2011), no. 1, Paper 22, 31. MR 2770127 (2012f:82016)
22. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their 𝑞-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096 (2011e:33029)
23. J. Merryfield, personal communication with the fourth author.
24. Philippe Nadeau, The structure of alternative tableaux, J. Combin. Theory Ser. A 118 (2011), no. 5, 1638–1660. MR 2771605 (2012j:05458), http://dx.doi.org/10.1016/j.jcta.2011.01.012
25. J.-C. Novelli, J.-Y. Thibon, and L. K. Williams, Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux, Adv. Math. 224 (2010), no. 4, 1311–1348. MR 2646299 (2011j:05362), http://dx.doi.org/10.1016/j.aim.2010.01.006
26. Jean-Guy Penaud, Une preuve bijective d’une formule de Touchard-Riordan, Discrete Math. 139 (1995), no. 1-3, 347–360 (French, with English and French summaries). Formal power series and algebraic combinatorics (Montreal, PQ, 1992). MR 1336847 (97b:05010), http://dx.doi.org/10.1016/0012-365X(94)00140-E
27. A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764v1, preprint (2006).
28. Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR 1442260 (98a:05001)
29. Einar Steingrímsson and Lauren K. Williams, Permutation tableaux and permutation patterns, J. Combin. Theory Ser. A 114 (2007), no. 2, 211–234. MR 2293088 (2008c:05004), http://dx.doi.org/10.1016/j.jcta.2006.04.001
30. Masaru Uchiyama, Tomohiro Sasamoto, and Miki Wadati, Asymmetric simple exclusion process with open boundaries and Askey-Wilson polynomials, J. Phys. A 37 (2004), no. 18, 4985–5002. MR 2065218 (2006d:82047), http://dx.doi.org/10.1088/0305-4470/37/18/006
31. X.G. Viennot, Une théorie combinatoire des polynômes orthogonaux, Notes de cours, UQÀM, Montréal, 1988.
32. X. Viennot, Slides from a talk at the Isaac Newton Institute, April 2008.
33. Lauren K. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005), no. 2, 319–342. MR 2102660 (2005i:05197), http://dx.doi.org/10.1016/j.aim.2004.01.003

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|