## Formulae for Askey-Wilson moments and enumeration of staircase tableaux

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- by S. Corteel, R. Stanley, D. Stanton and L. Williams PDF
- Trans. Amer. Math. Soc.
**364**(2012), 6009-6037 Request permission

## 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.

## References

- 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**, DOI 10.1007/BFb0086401 - 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**, DOI 10.1090/memo/0319 - J. C. Aval, A. Boussicault and S. Dasse-Hartaut, The tree structure in staircase tableaux, Gascom 2012. arxiv:1109.4907.
- R. Brak and J. W. Essam,
*Asymmetric exclusion model and weighted lattice paths*, J. Phys. A**37**(2004), no. 14, 4183–4217. MR**2066074**, DOI 10.1088/0305-4470/37/14/002 - Alexander Burstein,
*On some properties of permutation tableaux*, Ann. Comb.**11**(2007), no. 3-4, 355–368. MR**2376110**, DOI 10.1007/s00026-007-0323-0 - T. S. Chihara,
*An introduction to orthogonal polynomials*, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR**0481884** - Sylvie Corteel,
*Crossings and alignments of permutations*, Adv. in Appl. Math.**38**(2007), no. 2, 149–163. MR**2290808**, DOI 10.1016/j.aam.2006.01.006 - 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**, DOI 10.1016/j.aam.2010.04.009 - Sylvie Corteel and Philippe Nadeau,
*Bijections for permutation tableaux*, European J. Combin.**30**(2009), no. 1, 295–310. MR**2460235**, DOI 10.1016/j.ejc.2007.12.007 - 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**, DOI 10.1016/j.aam.2006.08.002 - 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.
- 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**, DOI 10.1073/pnas.0909915107 - 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**, DOI 10.1215/00127094-1433385 - 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** - B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier,
*Exact solution of a $1$D asymmetric exclusion model using a matrix formulation*, J. Phys. A**26**(1993), no. 7, 1493–1517. MR**1219679**, DOI 10.1088/0305-4470/26/7/011 - Enrica Duchi and Gilles Schaeffer,
*A combinatorial approach to jumping particles*, J. Combin. Theory Ser. A**110**(2005), no. 1, 1–29. MR**2128962**, DOI 10.1016/j.jcta.2004.09.006 - Philippe Flajolet,
*On congruences and continued fractions for some classical combinatorial quantities*, Discrete Math.**41**(1982), no. 2, 145–153. MR**676874**, DOI 10.1016/0012-365X(82)90201-1 - 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**, DOI 10.1016/0196-6774(80)90020-6 - 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**, DOI 10.1017/CBO9780511526251 - 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**, DOI 10.1016/S0021-9045(03)00076-5 - Matthieu Josuat-Vergès,
*Combinatorics of the three-parameter PASEP partition function*, Electron. J. Combin.**18**(2011), no. 1, Paper 22, 31. MR**2770127** - 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**, DOI 10.1007/978-3-642-05014-5 - J. Merryfield, personal communication with the fourth author.
- Philippe Nadeau,
*The structure of alternative tableaux*, J. Combin. Theory Ser. A**118**(2011), no. 5, 1638–1660. MR**2771605**, DOI 10.1016/j.jcta.2011.01.012 - 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**, DOI 10.1016/j.aim.2010.01.006 - 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**, DOI 10.1016/0012-365X(94)00140-E - A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764v1, preprint (2006).
- 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**, DOI 10.1017/CBO9780511805967 - Einar Steingrímsson and Lauren K. Williams,
*Permutation tableaux and permutation patterns*, J. Combin. Theory Ser. A**114**(2007), no. 2, 211–234. MR**2293088**, DOI 10.1016/j.jcta.2006.04.001 - 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**, DOI 10.1088/0305-4470/37/18/006 - X.G. Viennot, Une théorie combinatoire des polynômes orthogonaux, Notes de cours, UQÀM, Montréal, 1988.
- X. Viennot, Slides from a talk at the Isaac Newton Institute, April 2008.
- Lauren K. Williams,
*Enumeration of totally positive Grassmann cells*, Adv. Math.**190**(2005), no. 2, 319–342. MR**2102660**, DOI 10.1016/j.aim.2004.01.003

## Additional Information

**S. Corteel**- Affiliation: LIAFA, Centre National de la Recherche Scientifique et Université Paris Diderot, Paris 7, Case 7014, 75205 Paris Cedex 13 France
- MR Author ID: 633477
- Email: corteel@liafa.univ-paris-diderot.fr
**R. Stanley**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02138
- MR Author ID: 166285
- Email: rstan@math.mit.edu
**D. Stanton**- Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Email: stanton@math.umn.edu
**L. Williams**- Affiliation: Department of Mathematics, University of California, Berkeley, Evans Hall Room 913, Berkeley, California 94720
- MR Author ID: 611667
- Email: williams@math.berkeley.edu
- Received by editor(s): August 13, 2010
- Received by editor(s) in revised form: March 16, 2011
- Published electronically: May 2, 2012
- Additional Notes: The first author was partially supported by ANR grant ANR-08-JCJC-0011

The second author was partially supported by NSF grant No. 0604423

The fourth author was partially supported by NSF grant DMS-0854432 and an Alfred Sloan Fellowship. - © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**364**(2012), 6009-6037 - MSC (2010): Primary 05A15; Secondary 33C45, 82B23
- DOI: https://doi.org/10.1090/S0002-9947-2012-05588-7
- MathSciNet review: 2946941