%% Created for Torin Greenwood at 2023-06-09 @Misc{BJP:2023, AUTHOR = {Yuliy Baryshnikov and Kaitian Jin and Robin Pemantle}, TITLE = {Coefficient asymptotics of algebraic multivariable generating functions}, howpublished = {Preprint (online)}, URL = {https://ymb.web.illinois.edu/wp-content/uploads/2023/02/AlgebraicGF.pdf}, } @article{BeRi:1983, title = {Central and local limit theorems applied to asymptotic enumeration II: Multivariate generating functions}, journal = {Journal of Combinatorial Theory, Series A}, volume = {34}, number = {3}, pages = {255-265}, year = {1983}, issn = {0097-3165}, doi = {https://doi.org/10.1016/0097-3165(83)90062-6}, url = {https://www.sciencedirect.com/science/article/pii/0097316583900626}, author = {Edward A Bender and L {Bruce Richmond}}, abstract = {Let a multivariate sequence an(k) ⩾ 0 be given. Multivariate central and local limit theorems are proved for an(k) as n → ∞ that are based on examining the generating function. Applications are made to permutations with rises and falls, ordered partitions of sets, Tutte polynomials of recursive families, and dissections of polygons.} } @article{BMMi:2010, title={Walks with small steps in the quarter plane}, author={Bousquet-M{\'e}lou, M. and Mishna, M.}, journal={Contemporary Mathematics}, volume={520}, pages={1--40}, year={2010}} @article {Chu:2019, AUTHOR = {Chu, Wenchang}, TITLE = {Logarithms of a binomial series: extension of a series of {K}nuth}, JOURNAL = {Math. Commun.}, FJOURNAL = {Mathematical Communications}, VOLUME = {24}, YEAR = {2019}, NUMBER = {1}, PAGES = {83--90}, ISSN = {1331-0623,1848-8013}, MRCLASS = {05A15 (05A10)}, MRNUMBER = {3884556}, MRREVIEWER = {Tri\ Lai}, } @article{Dr:1994, title = {Asymptotic distributions and a multivariate darboux method in enumeration problems}, journal = {Journal of Combinatorial Theory, Series A}, volume = {67}, number = {2}, pages = {169-184}, year = {1994}, issn = {0097-3165}, doi = {https://doi.org/10.1016/0097-3165(94)90011-6}, url = {https://www.sciencedirect.com/science/article/pii/0097316594900116}, author = {Michael Drmota}, abstract = {Let c(x, z) = Σ cnkxnzk (cnk ⩾ 0) be a bivariate generating function satisfying a functional equation c = G(c, x, z). By using a central limit theorem of Bender it is shown that discrete random variables Xn with P[Xn = k] = cnk/(Σ cni) are asymptotically normal with mean μn ∼ μn and variance σn2 ∼ σ2n. Furthermore a bivariate asymptotic expansion for the coefficients cnk can be obtained by two different methods. After some applications to tree enumeration problems a multivariate Darboux-method is formulated.} } @article{FlOd:1990, Author = {Flajolet, Philippe and Odlyzko, Andrew}, Coden = {SJDMEC}, Fjournal = {SIAM Journal on Discrete Mathematics}, Issn = {0895-4801}, Journal = {SIAM J. Discrete Math.}, Mrclass = {05A15 (30E20 40E05 41A60)}, Mrnumber = {MR1039294 (90m:05012)}, Mrreviewer = {E. Rodney Canfield}, Number = {2}, Pages = {216--240}, Title = {Singularity analysis of generating functions}, Volume = {3}, Year = {1990}, URL = {https://doi.org/10.1137/0403019}} @book{FlSe:2009, Author = {Flajolet, Phillipe and Sedgewick, Robert}, Isbn = {0521898064}, Pages = {824}, Publisher = {Cambridge University Press}, Title = {Analytic combinatorics}, Year = {2009}} @article{Gao:1992, title = {Central and local limit theorems applied to asymptotic enumeration IV: multivariate generating functions}, journal = {Journal of Computational and Applied Mathematics}, volume = {41}, number = {1}, pages = {177-186}, year = {1992}, issn = {0377-0427}, doi = {https://doi.org/10.1016/0377-0427(92)90247-U}, url = {https://www.sciencedirect.com/science/article/pii/037704279290247U}, author = {Zhicheng Gao and L.Bruce Richmond}, keywords = {Generating function, limit theorem, asymptotic enumeration}, abstract = {Flajolet and Soria (1989, 1990) discussed some general combinatorial structures in which central limit theorem and exponential tail results hold. In this paper, we shall use Flajolet and Odlyzko's “transfer theorems” (1990) to extend Bender and Richmond's (1983) central and local limit theorems to a wider class of generating functions which will cover the above-mentioned combinatorial structures. The local limit theorem provides more accurate asymptotic information and implies the superexponential tail results.} } @article {GeZe:1992, AUTHOR = {Gessel, I. M. and Zeilberger, D.}, TITLE = {Random walks in a {W}eyl chamber}, JOURNAL = {Proceedings of the American Mathematical Society}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {115}, YEAR = {1992}, NUMBER = {1}, PAGES = {27--31}, MRCLASS = {05A15}, MRNUMBER = {1092920}, MRREVIEWER = {Sri Gopal Mohanty}, } @article{Greenwood:2018, title = {Asymptotics of bivariate analytic functions with algebraic singularities}, journal = {Journal of Combinatorial Theory, Series A}, volume = {153}, pages = {1-30}, year = {2018}, issn = {0097-3165}, doi = {https://doi.org/10.1016/j.jcta.2017.06.014}, url = {https://www.sciencedirect.com/science/article/pii/S0097316517300833}, author = {Torin Greenwood}, keywords = {Generating functions, Coefficients, Asymptotics, Multivariate, Singularity analysis, Algebraic}, abstract = {In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to derive asymptotic formulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Then, we apply these results to a generating function encoding information about the stationary distributions of a graph coloring algorithm studied by Butler, Chung, Cummings, and Graham (2015). Historically, Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions. These multivariate techniques are used here to analyze functions with algebraic singularities.} } @article{GMRW:2022, Author = {Torin Greenwood and Tiadora Ruza and Stephen Melczer and Mark C. Wilson}, Date-Added = {2022-09-21 22:49:49 -0500}, Date-Modified = {2022-09-21 22:52:02 -0500}, Journal = {S\'{e}minaire Lotharingien de Combinatoire (Proceedings of FPSAC 2022)}, Pages = {12}, Title = {Asymptotics of Coefficients of Algebraic Series Via Embedding Into Rational Series (Extended Abstract)}, Volume = {86B}, Year = {2022}, URL = {https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2022/30.pdf}} @article {Hackl:2018, AUTHOR = {Hackl, Benjamin and Prodinger, Helmut}, TITLE = {The necklace process: a generating function approach}, JOURNAL = {Statist. Probab. Lett.}, FJOURNAL = {Statistics \& Probability Letters}, VOLUME = {142}, YEAR = {2018}, PAGES = {57--61}, ISSN = {0167-7152}, MRCLASS = {60C05 (05A15 05A16)}, MRNUMBER = {3842616}, MRREVIEWER = {Adam Hammett}, DOI = {10.1016/j.spl.2018.06.010}, URL = {https://doi.org/10.1016/j.spl.2018.06.010}, } @article{GaRo:1992, title = {Central and local limit theorems applied to asymptotic enumeration IV: multivariate generating functions}, journal = {Journal of Computational and Applied Mathematics}, volume = {41}, number = {1}, pages = {177-186}, year = {1992}, issn = {0377-0427}, doi = {https://doi.org/10.1016/0377-0427(92)90247-U}, url = {https://www.sciencedirect.com/science/article/pii/037704279290247U}, author = {Zhicheng Gao and L.Bruce Richmond}, keywords = {Generating function, limit theorem, asymptotic enumeration}, abstract = {Flajolet and Soria (1989, 1990) discussed some general combinatorial structures in which central limit theorem and exponential tail results hold. In this paper, we shall use Flajolet and Odlyzko's “transfer theorems” (1990) to extend Bender and Richmond's (1983) central and local limit theorems to a wider class of generating functions which will cover the above-mentioned combinatorial structures. The local limit theorem provides more accurate asymptotic information and implies the superexponential tail results.} } @misc{Hwang:2022, title={Analysis of some exactly solvable diminishing urn models}, author={Hsien-Kuei Hwang and Markus Kuba and Alois Panholzer}, year={2022}, eprint={2212.05091}, archivePrefix={arXiv}, primaryClass={math.CO} } @misc{JaKo:2023, title={Logarithms of Catalan generating functions: A combinatorial approach}, author={Sabine Jansen and Leonid Kolesnikov}, year={2023}, eprint={2302.09661}, archivePrefix={arXiv}, primaryClass={math.CO} } @misc{Knuth:2014, author = {Knuth, D.E.}, title = {3/2-ary trees}, howpublished = {Annual Christmas Tree lecture}, month = {12}, year = {2014}, url = {https://www.youtube.com/watch?v=P4AaGQIo0HY} } @article {Knuth:2015, AUTHOR = {Knuth, D.E.}, TITLE = {Log–squared of the {C}atalan generating function}, JOURNAL = {Amer. Math. Monthly}, VOLUME = {122}, YEAR = {2015}, PAGES = {390} } @book{MePeWi:2024, place={Cambridge}, edition={2}, series={Cambridge Studies in Advanced Mathematics}, title={Analytic Combinatorics in Several Variables}, publisher={Cambridge University Press}, author={Pemantle, Robin and Wilson, Mark C. and Melczer, Stephen}, year={2024}, collection={Cambridge Studies in Advanced Mathematics}} @book{PeWi:2013, address = {New York}, author = {Pemantle, Robin and Wilson, Mark C.}, date-added = {2022-06-28 15:06:33 -0500}, date-modified = {2022-06-28 15:06:33 -0500}, isbn = {978-1107031579}, publisher = {Cambridge University Press}, title = {Analytic Combinatorics in Several Variables}, year = {2013}} @article {Wilf:1982, AUTHOR = {Wilf, Herbert S.}, TITLE = {What is an answer?}, JOURNAL = {Amer. Math. Monthly}, FJOURNAL = {American Mathematical Monthly}, VOLUME = {89}, YEAR = {1982}, NUMBER = {5}, PAGES = {289--292}, ISSN = {0002-9890,1930-0972}, MRCLASS = {05A15 (68C25)}, MRNUMBER = {653502}, DOI = {10.2307/2321713}, URL = {https://doi.org/10.2307/2321713}, }