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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
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
Published electronically: May 2, 2012
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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.


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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
Email: corteel@liafa.univ-paris-diderot.fr

R. Stanley
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02138
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
Email: williams@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05588-7
PII: S 0002-9947(2012)05588-7
Keywords: Staircase tableaux, asymmetric exclusion process, Askey-Wilson polynomials, permutations, matchings
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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.