{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T22:26:39Z","timestamp":1787005599729,"version":"build-2736575974"},"reference-count":22,"publisher":"National Academy of Sciences","issue":"15","content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2010,4,13]]},"abstract":"<jats:p>\n                    We introduce some combinatorial objects called staircase tableaux, which have cardinality 4\n                    <jats:sup>\n                      <jats:italic>n<\/jats:italic>\n                    <\/jats:sup>\n                    <jats:italic>n<\/jats:italic>\n                    \u00a0!, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics introduced in the late 1960s, which describes a system of interacting particles hopping left and right on a one-dimensional lattice of\n                    <jats:italic>n<\/jats:italic>\n                    sites with open boundaries. It has been cited as a model for traffic flow and translation in protein synthesis. In its most general form, particles may enter and exit at the left with probabilities \u03b1 and \u03b3, and they may exit and enter at the right with probabilities \u03b2 and \u03b4. In the bulk, the probability of hopping left is\n                    <jats:italic>q<\/jats:italic>\n                    times the probability of hopping right. Our first result is a formula for the stationary distribution of the ASEP with all parameters general, in terms of staircase tableaux. Our second result is a formula for the moments of (the weight function of) Askey-Wilson polynomials, also in terms of staircase tableaux. Since the 1980s there has been a great deal of work giving combinatorial formulas for moments of classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre); among these polynomials, the Askey-Wilson polynomials are the most important, because they are at the top of the hierarchy of classical orthogonal polynomials.\n                  <\/jats:p>","DOI":"10.1073\/pnas.0909915107","type":"journal-article","created":{"date-parts":[[2010,3,26]],"date-time":"2010-03-26T23:35:48Z","timestamp":1269646548000},"page":"6726-6730","update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":22,"title":["Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials"],"prefix":"10.1073","volume":"107","author":[{"given":"Sylvie","family":"Corteel","sequence":"first","affiliation":[{"name":"Laboratoire d\u2019Informatique Algorithmique: Fondements et Applications, Centre National de la Recherche Scientifique et Universit\u00e9 Paris Diderot\u2013Paris 7, Case 7014, 75205, Paris Cedex 13, France; and"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lauren K.","family":"Williams","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, Berkeley, CA 94720"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2010,3,26]]},"reference":[{"key":"e_1_3_3_1_2","doi-asserted-by":"publisher","DOI":"10.1002\/bip.1968.360060102"},{"key":"e_1_3_3_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(70)90034-4"},{"key":"e_1_3_3_3_2","volume-title":"Traffic and Granular Flow \u201997","author":"Schreckenberg M","year":"1998","unstructured":"M Schreckenberg, D Wolf Traffic and Granular Flow \u201997 (Springer, Singapore, 1998)."},{"key":"e_1_3_3_4_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.65.031911"},{"key":"e_1_3_3_5_2","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/32\/41\/306"},{"key":"e_1_3_3_6_2","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/37\/18\/006"},{"key":"e_1_3_3_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2004.09.006"},{"key":"e_1_3_3_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2006.01.006"},{"key":"e_1_3_3_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00275693"},{"key":"e_1_3_3_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2006.08.002"},{"key":"e_1_3_3_11_2","first-page":"mm055","article-title":"A Markov chain on permutations which projects to the asymmetric exclusion process","author":"Corteel S","year":"2007","unstructured":"S Corteel, L Williams, A Markov chain on permutations which projects to the asymmetric exclusion process. Int Math Res Notices, pp. mm055 (2007).","journal-title":"Int Math Res Notices"},{"key":"e_1_3_3_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2006.04.001"},{"key":"e_1_3_3_13_2","unstructured":"A Postnikov Total positivity Grassmannians and networks. arXiv:math\/0609764v1. (2006)."},{"key":"e_1_3_3_14_2","first-page":"1","article-title":"Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials","volume":"54","author":"Askey R","year":"1985","unstructured":"R Askey, J Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. 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