An elementary solution of Gessel's walks in the quadrant

Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N2, starting and ending at the origin (0,0) and taking their steps in {→,↗,←,↙} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (t...

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Published in:	Advances in mathematics (New York. 1965) Vol. 303; pp. 1171 - 1189
Main Author:	Bousquet-Mélou, Mireille
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 05.11.2016 Elsevier
Subjects:	Algebraic series Combinatorics Exact enumeration Lattice walks Mathematics Exact enumeration Algebraic series Lattice walks
ISSN:	0001-8708, 1090-2082
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Abstract	Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N2, starting and ending at the origin (0,0) and taking their steps in {→,↗,←,↙} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is, steps in {−1,0,1}2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations. A complete classification of quadrant walks according to the nature of their generating function (algebraic, D-finite or not) is now available, but Gessel's walks remained mysterious because they were the only model among the 23 D-finite ones that had not been given an elementary solution. Instead, Gessel's conjecture was first proved using an inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.
AbstractList	Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N2, starting and ending at the origin (0,0) and taking their steps in {→,↗,←,↙} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is, steps in {−1,0,1}2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations. A complete classification of quadrant walks according to the nature of their generating function (algebraic, D-finite or not) is now available, but Gessel's walks remained mysterious because they were the only model among the 23 D-finite ones that had not been given an elementary solution. Instead, Gessel's conjecture was first proved using an inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra. Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in {E,NE,W,SW} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is,steps in {-1,0,1}^2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations.A complete classification of quadrant walks according to the nature of their generating function(algebraic, D-finite or not) is now available, but Gessel'swalks remained mysterious because they were the only model among the 23D-finite ones that had not been given an elementarysolution. Instead, Gessel's conjecture was first proved usingan inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.
Author	Bousquet-Mélou, Mireille
Author_xml	– sequence: 1 givenname: Mireille surname: Bousquet-Mélou fullname: Bousquet-Mélou, Mireille email: bousquet@labri.fr organization: CNRS, LaBRI, Université de Bordeaux, 351 cours de la Libération, F-33405 Talence Cedex, France
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Cites_doi	10.1016/j.disc.2012.09.003 10.1016/j.tcs.2009.04.008 10.1016/S0012-365X(00)00147-3 10.1016/j.crma.2014.11.015 10.1080/10236190802332084 10.4171/jems/317 10.1214/105051605000000052 10.1090/tran/6804 10.1016/j.spa.2013.12.003 10.1016/S0304-3975(03)00219-6 10.1016/j.jctb.2005.12.003 10.1016/j.aam.2010.11.004 10.1016/j.jcta.2013.09.005 10.4153/CJM-1964-058-7 10.1007/BF01370732 10.1016/j.jctb.2011.02.003 10.1137/0144074 10.1090/S0002-9939-2010-10398-2 10.1016/j.jcta.2008.06.011 10.4153/CJM-1962-002-9 10.1016/S0304-3975(02)00007-5 10.1007/s00026-016-0328-7 10.1017/S0963548314000145 10.1016/0378-3758(86)90009-1 10.1017/S0027763000022698 10.1137/S0895480190177650 10.1073/pnas.0901678106 10.37236/1271 10.1007/BF01845637 10.1007/s10801-010-0259-z 10.1016/S0012-365X(01)00250-3 10.1007/s10240-012-0045-7
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Snippet	Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N2, starting and ending at the origin (0,0) and taking their steps in... Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in...
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Title	An elementary solution of Gessel's walks in the quadrant
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