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@incollection{kauers2015ore,
Author = {Kauers, M. and Jaroschek, M. and Johansson, F.},
Booktitle = {Computer Algebra and Polynomials},
Date-Added = {2018-11-05 10:20:15 +1100},
Date-Modified = {2018-11-05 10:21:22 +1100},
Doi = {10.1007/978-3-319-15081-9_6},
Pages = {105--125},
Publisher = {Springer},
Title = {Ore polynomials in {S}age},
Year = {2015},
Bdsk-Url-1 = {https://doi.org/10.1007/978-3-319-15081-9_6}}
@article{raschel2012,
Author = {Raschel, K.},
Date-Added = {2018-11-05 09:21:37 +1100},
Date-Modified = {2018-11-05 09:22:22 +1100},
Doi = {10.4171/jems/317},
Journal = {J. Europ. Math. Soc.},
Number = {3},
Pages = {749--777},
Title = {Counting walks in a quadrant: a unified approach via boundary value problems},
Volume = {14},
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.4171/jems/317}}
@inproceedings{bostan2009automatic,
Author = {Bostan, A. and Kauers, M.},
Booktitle = {Proceedings of FPSAC'09},
Date-Added = {2018-11-05 09:15:54 +1100},
Date-Modified = {2019-05-09 22:40:53 +1000},
Pages = {201--215},
Publisher = {Discrete Math. Theor. Comput. Sci.},
Title = {Automatic classification of restricted lattice walks},
Url = {https://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAK0117.1.html},
Year = {2009},
Bdsk-Url-1 = {https://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAK0117.1.html}}
@book{fayolle1999random,
Author = {Fayolle, G. and Malyshev, V.A. and Iasnogorodski, R.},
Date-Added = {2018-11-05 09:13:16 +1100},
Date-Modified = {2018-11-05 09:14:03 +1100},
Doi = {10.1007/978-3-642-60001-2},
Publisher = {Springer},
Title = {{Random Walks in the Quarter-Plane}},
Year = {1999},
Bdsk-Url-1 = {https://doi.org/10.1007/978-3-642-60001-2}}
@article{tabbara_exact_2014,
Abstract = {We find the exact solution of two interacting friendly directed walks (modelling polymers) on the square lattice. These walks are confined to the quarter plane by a horizontal attractive surface, to capture the effects of DNA-denaturation and adsorption. We find the solution to the model's corresponding generating function by means of the obstinate kernel method . Specifically, we apply this technique in two different instances to establish partial solutions for two simplified generating functions of the same underlying model that ignore either surface or shared contacts. We then subsequently combine our two partial solutions to find the solution for the full generating function in terms of the two simpler variants. This expression guides our analysis of the model, where we find the system exhibits four phases, and proceed to delineate the full phase diagram, showing that all observed phase transitions are second-order.},
Author = {Tabbara, R. and Owczarek, A. L. and Rechnitzer, A.},
Date-Added = {2018-11-02 17:06:53 +1100},
Date-Modified = {2018-11-02 17:08:49 +1100},
Doi = {10.1088/1751-8113/47/1/015202},
Journal = {J. Phys. A: Math. Theor.},
Number = {1},
Pages = {015202},
Title = {An exact solution of two friendly interacting directed walks near a sticky wall},
Volume = {47},
Year = {2014},
Bdsk-Url-1 = {http://stacks.iop.org/1751-8121/47/i=1/a=015202},
Bdsk-Url-2 = {https://doi.org/10.1088/1751-8113/47/1/015202}}
@article{mishna2009classifying,
Abstract = {This work considers the nature of generating functions of random lattice walks restricted to the first quadrant. In particular, we find combinatorial criteria to decide if related series are algebraic, transcendental holonomic or otherwise. Complete results for walks taking their steps in a maximum of three directions of restricted amplitude are given, as is a well-supported conjecture for all walks with steps taken from a subset of \{0,$\pm$1\}2. New enumerative results are presented for several classes, each obtained with a variant of the kernel method.},
Author = {Mishna, M.},
Date-Added = {2018-11-02 17:05:55 +1100},
Date-Modified = {2018-11-02 17:06:22 +1100},
Doi = {10.1016/j.jcta.2008.06.011},
Journal = {J. Combin. Theory Ser. A},
Number = {2},
Pages = {460--477},
Title = {Classifying lattice walks restricted to the quarter plane},
Volume = {116},
Year = {2009},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0097316508001179},
Bdsk-Url-2 = {https://doi.org/10.1016/j.jcta.2008.06.011}}
@article{bousquet2005walks,
Abstract = {We consider planar lattice walks that start from (0,0), remain in the first quadrant i,j≥0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: * they are counted by nice numbers [Kreweras, Cahiers du B.U.R.O 6 (1965) 5--105], * the generating function of these numbers is algebraic [Gessel, J. Statist. Plann. Inference 14 (1986) 49--58], * the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function [Flatto and Hahn, SIAM J. Appl. Math. 44 (1984) 1041--1053]. These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, which is more elementary that those previously published. We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds. Our proofs are based on the solution of certain functional equations, which are very simple to establish. Finding purely combinatorial proofs remains an open problem.},
Author = {Bousquet-M{\'e}lou, M.},
Date-Added = {2018-11-02 17:01:57 +1100},
Date-Modified = {2018-11-02 17:02:51 +1100},
Doi = {10.1214/105051605000000052},
Journal = {Ann. Appl. Prob.},
Number = {2},
Pages = {1451--1491},
Title = {Walks in the quarter plane: {Kreweras}' algebraic model},
Volume = {15},
Year = {2005},
Bdsk-Url-1 = {https://projecteuclid.org/euclid.aoap/1115137982},
Bdsk-Url-2 = {https://doi.org/10.1214/105051605000000052}}
@incollection{bousquet2010walks,
Abstract = {Let S be a subset of -1,0,1{\textasciicircum}2 not containing (0,0). We address the enumeration of plane lattice walks with steps in S, that start from (0,0) and always remain in the first quadrant. A priori, there are 2{\textasciicircum}8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a half-plane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study. To each of them, we associate a group G of birational transformations. We show that this group is finite in exactly 23 cases. We present a unified way of solving 22 of the 23 models associated with a finite group. For each of them, the generating function is found to be D-finite. The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al. to have an algebraic (and hence D-finite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a non-D-finite generating function. Our approach allows us to recover and refine some known results, and also to obtain new results. For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function.},
Author = {Bousquet-M{\'e}lou, M. and Mishna, M.},
Booktitle = {Algorithmic Probability and Combinatorics},
Date-Added = {2018-11-02 16:58:42 +1100},
Date-Modified = {2019-05-09 22:38:35 +1000},
Doi = {10.1090/conm/520},
Number = {520},
Pages = {1--40},
Publisher = {AMS},
Series = {Contemporary Mathematics},
Title = {Walks with small steps in the quarter plane},
Year = {2010},
Bdsk-Url-1 = {https://doi.org/10.1090/conm/520}}
@article{flajolet1980combinatorial,
Author = {Flajolet, Philippe},
File = {:C\:\\xrj\\ordered files\\pub\\method\\combinatorial\\combinatorial aspects of continued fractions.pdf:PDF},
Journal = {Discrete Mathematics},
Number = {2},
Pages = {125--161},
Publisher = {Elsevier},
Title = {Combinatorial aspects of continued fractions},
Volume = {32},
Year = {1980}}
@article{gessel1985binomial,
Author = {Gessel, Ira and Viennot, G{\'e}rard},
File = {:C\:\\xrj\\ordered files\\pub\\method\\gessel-viennot_binomial determinants, paths, and hook length formulae_1985_398657.pdf:PDF},
Journal = {Advances in mathematics},
Number = {3},
Pages = {300--321},
Publisher = {Elsevier},
Title = {Binomial determinants, paths, and hook length formulae},
Volume = {58},
Year = {1985}}
@article{lubensky2000pulling,
Author = {Lubensky, David K and Nelson, David R},
Journal = {Physical review letters},
Number = {7},
Pages = {1572},
Publisher = {APS},
Title = {Pulling pinned polymers and unzipping DNA},
Volume = {85},
Year = {2000}}
@article{dulucq1998baxter,
Author = {Dulucq, Serge and Guibert, Olivier},
File = {:C\:\\xrj\\ordered files\\pub\\method\\combinatorial\\Baxter permutations .pdf:PDF},
Journal = {Discrete Mathematics},
Number = {1-3},
Pages = {143--156},
Publisher = {Elsevier},
Title = {Baxter permutations},
Volume = {180},
Year = {1998}}
@article{blythe2007nonequilibrium,
Author = {Blythe, Richard A and Evans, Martin R},
File = {:C\:\\xrj\\ordered files\\subjects\\Solvable Model\\BlytheEvans.pdf:PDF},
Journal = {Journal of Physics A: Mathematical and Theoretical},
Number = {46},
Pages = {R333},
Publisher = {IOP Publishing},
Title = {Nonequilibrium steady states of matrix-product form: a solver's guide},
Volume = {40},
Year = {2007}}
@article{elizalde2012clusters,
Author = {Elizalde, Sergi and Noy, Marc},
File = {:C\:\\xrj\\ordered files\\pub\\method\\combinatorial\\Clusters, generating functions and asymptotics for consecutive patterns in permutations.pdf:PDF},
Journal = {Advances in Applied Mathematics},
Number = {3-5},
Pages = {351--374},
Publisher = {Elsevier},
Title = {Clusters, generating functions and asymptotics for consecutive patterns in permutations},
Volume = {49},
Year = {2012}}
@article{bousquet2002counting,
Author = {Bousquet-M{\'e}lou, Mireille},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\COUNTING WALKS IN THE QUARTER PLANE.pdf:PDF},
Journal = {Mathematics and computer science},
Pages = {49--67},
Title = {Counting walks in the quarter plane},
Volume = {2},
Year = {2002}}
@article{bernardi2007bijective,
Author = {Bernardi, Olivier},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\Bijective counting of Kreweras walks and loopless triangulations.pdf:PDF},
Journal = {Journal of Combinatorial Theory, Series A},
Number = {5},
Pages = {931--956},
Publisher = {Elsevier},
Title = {Bijective counting of Kreweras walks and loopless triangulations},
Volume = {114},
Year = {2007}}
@article{banderier2002basic,
Author = {Banderier, Cyril and Flajolet, Philippe},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\Basic analytic combinatorics of directed lattice paths.pdf:PDF},
Journal = {Theoretical Computer Science},
Number = {1-2},
Pages = {37--80},
Publisher = {Elsevier},
Title = {Basic analytic combinatorics of directed lattice paths},
Volume = {281},
Year = {2002}}
@article{brak2005directed,
Author = {Brak, Richard and Owczarek, Aleksander L and Rechnitzer, Andrew and Whittington, Stuart G},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\A directed walk model of a long chain polymer in a slit with attractive walls.pdf:PDF},
Journal = {Journal of Physics A: Mathematical and General},
Number = {20},
Pages = {4309},
Publisher = {IOP Publishing},
Title = {A directed walk model of a long chain polymer in a slit with attractive walls},
Volume = {38},
Year = {2005}}
@article{tabbara2016exact,
Author = {Tabbara, R and Owczarek, AL and Rechnitzer, A},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\An exact solution of three interacting friendly walks in the bulk.pdf:PDF},
Journal = {Journal of Physics A: Mathematical and Theoretical},
Number = {15},
Pages = {154004},
Publisher = {IOP Publishing},
Title = {An exact solution of three interacting friendly walks in the bulk},
Volume = {49},
Year = {2016}}
@article{owczarek2010enumeration,
Author = {Owczarek, Aleksander L and Prellberg, Thomas},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\Enumeration of area-weighted Dyck paths with restricted height.pdf:PDF},
Journal = {arXiv preprint arXiv:1004.1699},
Title = {Enumeration of area-weighed Dyck paths with restricted height},
Year = {2010}}
@article{prodinger2004kernel,
Author = {Prodinger, Helmut},
File = {:C\:\\xrj\\ordered files\\pub\\method\\kernel method\\kernel method.pdf:PDF},
Journal = {S{\'e}m. Lothar. Combin},
Pages = {B50f},
Title = {The kernel method: a collection of examples},
Volume = {50},
Year = {2004}}
@article{bousquet2016square,
Author = {Bousquet-M{\'e}lou, Mireille},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\Square lattice walks avoiding a quadrant.pdf:PDF},
Journal = {Journal of Combinatorial Theory, Series A},
Pages = {37--79},
Publisher = {Elsevier},
Title = {Square lattice walks avoiding a quadrant},
Volume = {144},
Year = {2016}}
@article{flajolet1987analytic,
Author = {Flajolet, Philippe},
File = {:C\:\\xrj\\ordered files\\pub\\method\\combinatorial\\Analytic models and ambiguity of context-free languages.pdf:PDF},
Journal = {Theoretical Computer Science},
Number = {2-3},
Pages = {283--309},
Publisher = {Elsevier},
Title = {Analytic models and ambiguity of context-free languages},
Volume = {49},
Year = {1987}}
@article{wilf12generatingfunctionology,
Author = {Wilf, H},
File = {:C\:\\xrj\\mathmast\\generatingfunctionology.pdf:PDF},
Journal = {ISBN: 0-12-751956-4},
Title = {Generatingfunctionology,(1990)}}
@article{bousquet2003four,
Author = {Bousquet-M{\'e}lou, Mireille},
File = {:C\:\\xrj\\ordered files\\pub\\method\\combinatorial\\Four classes of pattern-avoiding permutations under one roof generating trees with two labels.pdf:PDF},
Journal = {the electronic journal of combinatorics},
Number = {2},
Pages = {R19},
Title = {Four classes of pattern-avoiding permutations under one roof: generating trees with two labels},
Volume = {9},
Year = {2003}}
@article{mishna2009two,
Author = {Mishna, Marni and Rechnitzer, Andrew},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\Two non-holonomic lattice walks in the quarter plane.pdf:PDF},
Journal = {Theoretical Computer Science},
Number = {38-40},
Pages = {3616--3630},
Publisher = {Elsevier},
Title = {Two non-holonomic lattice walks in the quarter plane},
Volume = {410},
Year = {2009}}
@book{charalambides2002enumerative,
Author = {Charalambides, Charalambos A},
File = {:C\:\\xrj\\ordered files\\pub\\method\\combinatorial\\Enumerative combinatorics.pdf:PDF},
Publisher = {CRC Press},
Title = {Enumerative combinatorics},
Year = {2002}}
@article{bostan2010complete,
Author = {Bostan, Alin and Kauers, Manuel},
File = {:C\:\\xrj\\ordered files\\pub\\lattice walk\\The complete generating fun tion for Gessel walks is algebraic.pdf:PDF},
Journal = {Proceedings of the American Mathematical Society},
Number = {9},
Pages = {3063--3078},
Title = {The complete generating function for Gessel walks is algebraic},
Volume = {138},
Year = {2010}}
@article{bousquet2006polynomial,
Author = {Bousquet-M{\'e}lou, Mireille and Jehanne, Arnaud},
File = {:C\:\\xrj\\ordered files\\pub\\method\\combinatorial\\Polynomial equations with one catalytic variable.pdf:PDF},
Journal = {Journal of Combinatorial Theory, Series B},
Number = {5},
Pages = {623--672},
Publisher = {Elsevier},
Title = {Polynomial equations with one catalytic variable, algebraic series and map enumeration},
Volume = {96},
Year = {2006}}
@article{gessel1980factorization,
Author = {Gessel, Ira M},
File = {:C\:\\xrj\\ordered files\\pub\\method\\combinatorial\\A Factorization for Formal Laurent Series and Lattice Path Enumeration.pdf:PDF},
Journal = {Journal of Combinatorial Theory, Series A},
Number = {3},
Pages = {321--337},
Publisher = {Elsevier},
Title = {A factorization for formal Laurent series and lattice path enumeration},
Volume = {28},
Year = {1980}}
@article{beaton2018exact,
Author = {Beaton, N. R. and Owczarek, A. L. and Rechnitzer, A.},
Date-Modified = {2020-02-05 14:44:17 +1100},
Doi = {10.37236/8024},
Journal = {Electr. J. Combin.},
Number = {3},
Pages = {Art. P3.53},
Title = {Exact solution of some quarter plane walks with interacting boundaries},
Volume = {26},
Year = {2019}}