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Formulae and Asymptotics for Coefficients of Algebraic Functions



We study the coefficients of algebraic functions ∑ n≥0 f n z n . First, we recall the too-little-known fact that these coefficients f n always admit a closed form. Then we study their asymptotics, known to be of the type f n ~ CA n n α. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values for A. We extend what Philippe Flajolet called the Drmota–Lalley–Woods theorem (which states that α=−3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviours in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar (i.e., their generating functions are not ℕ-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota–Lalley–Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for ℕ-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).



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[1] Abel, N. H. (1992) Œuvres Complètes, Vol. II, Éditions J. Gabay, Sceaux. Reprint of the second edition of 1881.
[2] Adamczewski, B. and Bell, J. P. (2012) On vanishing coefficients of algebraic power series over fields of positive characteristic. Inventio Math. 187 343393.
[3] Albert, M. H. and Atkinson, M. D. (2005) Simple permutations and pattern restricted permutations. Discrete Math. 300 115.
[4] Allouche, J.-P. and Shallit, J. (2003) Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press.
[5] Autebert, J.-M., Flajolet, P. and Gabarró, J. (1987) Prefixes of infinite words and ambiguous context-free languages. Inform. Process. Lett. 25 211216.
[6] Banderier, C. (2001) Combinatoire analytique des chemins et des cartes. PhD thesis, Université Paris-VI.
[7] Banderier, C. (2002) Limit laws for basic parameters of lattice paths with unbounded jumps. In Mathematics and Computer Science II: Algorithms, Trees, Combinatorics and Probabilities (Chauvin, B. et al., eds), Trends in Mathematics, Birkhäuser, pp. 3347.
[8] Banderier, C., Bodini, O., Ponty, Y. and Tafat, H. (2012) On the diversity of pattern distributions in combinatorial systems. In Proc. Analytic Algorithmics and Combinatorics: ANALCO'12, SIAM, pp. 107116.
[9] Banderier, C., Bousquet-Mélou, M., Denise, A., Flajolet, P., Gardy, G. and Gouyou-Beauchamps, D. (2002) Generating functions for generating trees. Discrete Math. 246 2955.
[10] Banderier, C. and Flajolet, P. (2002) Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci. 281 3780.
[11] Banderier, C., Flajolet, P., Schaeffer, G. and Soria, M. (2001) Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Struct. Alg. 19 194246.
[12] Banderier, C. and Gittenberger, B. (2006) Analytic combinatorics of lattice paths: Enumeration and asymptotics for the area. In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proc. AG, pp. 345355.
[13] Banderier, C. and Hitczenko, P. (2012) Enumeration and asymptotics of restricted compositions having the same number of parts. Discrete Appl. Math. 160 25422554.
[14] Banderier, C. and Merlini, D. (2002) Lattice paths with an infinite set of jumps. In Proc. FPSAC 2002.
[15] Banderier, C. and Schwer, S. (2005) Why Delannoy numbers? J. Statist. Plann. Inference 135 4054.
[16] Barcucci, E., Del Lungo, A., Frosini, A. and Rinaldi, S. (2001) A technology for reverse-engineering a combinatorial problem from a rational generating function. Adv. Appl. Math. 26 129153.
[17] Bassino, F., Bouvel, M., Pivoteau, C., Pierrot, A. and Rossin, D. (2012) Combinatorial specification of permutation classes. In Proc. FPSAC '2012, DMTCS Proc. AR, pp. 791802.
[18] Béal, M.-P., Blocklet, M. and Dima, C. (2014) Zeta functions of finite-type-Dyck shifts are ℕ-algebraic. In Proc. 2014 Information Theory and Applications Workshop, IEEE.
[19] Bell, J. P., Burris, S. N. and Yeats, K. (2012) On the set of zero coefficients of a function satisfying a linear differential equation. Math. Proc. Cambridge Philos. Soc. 153 235247.
[20] Bergeron, F., Labelle, G. and Leroux, P. (1998) Combinatorial Species and Tree-Like Structures, Vol. 67 of Encyclopedia of Mathematics and its Applications, Cambridge University Press. Translated from the 1994 French original by Margaret Readdy.
[21] Berstel, J. (1971) Sur les pôles et le quotient de Hadamard de séries ℕ-rationnelles. CR Acad. Sci. Paris Sér. A–B 272 A1079A1081.
[22] Berstel, J. and Boasson, L. (1996) Towards an algebraic theory of context-free languages. Fund. Inform. 25 217239.
[23] Berstel, J. and Reutenauer, C. (2011) Noncommutative Rational Series with Applications, Vol. 137 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.
[24] Bertoni, A., Goldwurm, M. and Santini, M. (2001) Random generation for finitely ambiguous context-free languages. Theor. Inform. Appl. 35 499512.
[25] Beukers, F. and Heckman, G. (1989) Monodromy for the hypergeometric function n F n–1 . Inventio Math. 95 325354.
[26] Bodini, O., Darrasse, A. and Soria, M. (2008) Distances in random Apollonian network structures. In 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2008, DMTCS Proc. AJ, pp. 307318.
[27] Bodini, O. and Ponty, Y. (2010) Multi-dimensional Boltzmann sampling of languages. In 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms: AofA'10, DMTCS Proc. AM, pp. 4964.
[28] Bostan, A., Chyzak, F., Lecerf, G., Salvy, B. and Schost, E. (2007) Differential equations for algebraic functions. In Proc. ISSAC 2007, ACM, pp. 2532.
[29] Bostan, A. and Kauers, M. (2009) Automatic classification of restricted lattice walks. In Proc. FPSAC '09 (Krattenthaler, C., Strehl, V. and Kauers, M., eds), pp. 201215.
[30] Bostan, A. and Kauers, M. (2010) The complete generating function for Gessel walks is algebraic. Proc. Amer. Math. Soc. 138 30633078.
[31] Bostan, A., Lairez, P. and Salvy, B. (2014) Integral representations of binomial sums. In preparation.
[32] Bousquet-Mélou, M. (2006) Rational and algebraic series in combinatorial enumeration. In Proc. International Congress of Mathematicians, Vol. III, European Mathematical Society, pp. 789826.
[33] Bousquet-Mélou, M. and Jehanne, A. (2006) Polynomial equations with one catalytic variable, algebraic series and map enumeration. J. Combin. Theory Ser. B 96 623672.
[34] Bousquet-Mélou, M. and Schaeffer, G. (2002) Walks on the slit plane. Probab. Theory Rel. Fields 124 305344.
[35] Canou, B. and Darrasse, A. (2009) Fast and sound random generation for automated testing and benchmarking in objective Caml. In Proc. 2009 ACM SIGPLAN Workshop on ML: ML'09, pp. 6170.
[36] Ceccherini-Silberstein, T. and Woess, W. (2002) Growth and ergodicity of context-free languages. Trans. Amer. Math. Soc. 354 45974625.
[37] Ceccherini-Silberstein, T. and Woess, W. (2003) Growth-sensitivity of context-free languages. Theoret. Comput. Sci. 307 103116.
[38] Ceccherini-Silberstein, T. and Woess, W. (2012) Context-free pairs of groups I: Context-free pairs and graphs. European J. Combin. 33 14491466.
[39] Chomsky, N. and Schützenberger, M.-P. (1963) The algebraic theory of context-free languages. Computer Programming and Formal Systems, Vol. 26 of Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 118161.
[40] Christol, G., Kamae, T., Mendès France, M. and Rauzy, G. (1980) Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108 401419.
[41] Chudnovsky, D. V. and Chudnovsky, G. V. (1987) On expansion of algebraic functions in power and Puiseux series II. J. Complexity 3 125.
[42] Clote, P., Ponty, Y. and Steyaert, J.-M. (2012) Expected distance between terminal nucleotides of RNA secondary structures. J. Math. Biol. 65 581599.
[43] Cockle, J. (1861) On transcendental and algebraic solution. Philos. Mag. XXI 379383.
[44] Comtet, L. (1964) Calcul pratique des coefficients de Taylor d'une fonction algébrique. Enseignement Math. (2) 10 267270.
[45] Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel. Enlarged edition of the two volumes Analyse Combinatoire published in French in 1970 by Presses Universitaires de France.
[46] Darrasse, A. (2008) Random XML sampling the Boltzmann way. arXiv:0807.0992v1
[47] Delest, M. (1996) Algebraic languages: A bridge between combinatorics and computer science. In Formal Power Series and Algebraic Combinatorics 1994, Vol. 24 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, AMS, pp. 7187.
[48] Delest, M.-P. and Viennot, G. (1984) Algebraic languages and polyominoes enumeration. Theoret. Comput. Sci. 34 169206.
[49] Denef, J. and Lipshitz, L. (1984) Power series solutions of algebraic differential equations. Math. Ann. 267 213238.
[50] Denise, A., Ponty, Y. and Termier, M. (2003) Random generation of structured genomic sequences (poster). RECOMB'2003, Berlin, April 2003.
[51] Denise, A. and Zimmermann, P. (1999) Uniform random generation of decomposable structures using floating-point arithmetic. Theoret. Comput. Sci. 218 233248.
[52] Drmota, M. (1994) Asymptotic distributions and a multivariate Darboux method in enumeration problems. J. Combin. Theory Ser. A 67 169184.
[53] Drmota, M. (1997) Systems of functional equations. Random Struct. Alg. 10 103124.
[54] Drmota, M. (2009) Random Trees: An Interplay Between Combinatorics and Probability, Springer.
[55] Drmota, M., Gittenberger, B. and Morgenbesser, J. F. (2012) Infinite systems of functional equations and Gaussian limiting distributions. In 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms: AofA'12, DMTCS Proc. AQ, pp. 453478.
[56] Drmota, M. and Soria, M. (1995) Marking in combinatorial constructions: Generating functions and limiting distributions. Theor. Comput. Sci. 144 6799.
[57] Droste, M., Kuich, W. and Vogler, H., eds (2009) Handbook of Weighted Automata, Monographs in Theoretical Computer Science, Springer.
[58] Duchon, P. (1999) q-grammars and wall polyominoes. Ann. Combin. 3 311321.
[59] Duchon, P. (2000) On the enumeration and generation of generalized Dyck words. Discrete Math. 225 121135.
[60] Duchon, P., Flajolet, P., Louchard, G. and Schaeffer, G. (2004) Boltzmann samplers for the random generation of combinatorial structures. Combin. Probab. Comput. 13 577625.
[61] Flajolet, P. (1987) Analytic models and ambiguity of context-free languages. Theoret. Comput. Sci. 49 283309.
[62] Flajolet, P. and Noy, M. (1999) Analytic combinatorics of non-crossing configurations. Discrete Math. 204 203229.
[63] Flajolet, P. and Odlyzko, A. (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216240.
[64] Flajolet, P., Pelletier, M. and Soria, M. (2011) On Buffon machines and numbers. In Proc. Twenty-Second Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, pp. 172183.
[65] Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.
[66] Flajolet, P., Zimmerman, P. and Van Cutsem, B. (1994) A calculus for the random generation of labelled combinatorial structures. Theoret. Comput. Sci. 132 135.
[67] Furstenberg, H. (1967) Algebraic functions over finite fields. J. Algebra 7 271277.
[68] Goulden, I. P and Jackson, D. M. (2004) Combinatorial Enumeration, Dover. Reprint of the 1983 original.
[69] Grigorchuk, R. and de la Harpe, P. (1997) On problems related to growth, entropy, and spectrum in group theory. J. Dynam. Control Systems 3 5189.
[70] Harley, R. (1862) On the theory of the transcendental solution of algebraic equations. Quart. J. Pure Appl. Math. 5 337361.
[71] Harris, W. A. Jr and Sibuya, Y. (1985) The reciprocals of solutions of linear ordinary differential equations. Adv. Math. 58 119132.
[72] Hickey, T. and Cohen, J. (1983) Uniform random generation of strings in a context-free language. SIAM J. Comput. 12 645655.
[73] Joyal, A. (1981) Une théorie combinatoire des séries formelles. Adv. Math. 42 182.
[74] Jungen, R. (1931) Sur les séries de Taylor n'ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence. Comment. Math. Helv. 3 266306.
[75] Kauers, M., Jaroschek, M. and Johansson, F. (2014) Ore polynomials in Sage. arXiv:1306.4263 In: Computer Algebra and Polynomials, Jaime Gutierrez, Josef Schicho, Martin Weimann (ed.), Lecture Notes in Computer Science, to appear.
[76] Kauers, M. and Pillwein, V. (2010) When can we decide that a p-finite sequence is positive? In Proc. ISSAC'10, pp. 195–202. arXiv:1005.0600
[77] Kemp, R. (1980) A note on the density of inherently ambiguous context-free languages. Acta Inform. 14 295298.
[78] Kemp, R. (1993) Random multidimensional binary trees. J. Information Processing and Cybernetics (Elektron. Inform. Kybernet.) 29 936.
[79] Kleene, S. C. (1956) Representation of events in nerve nets and finite automata. In Automata Studies, Vol. 34 of Annals of Mathematics Studies, Princeton University Press, pp. 341.
[80] Knuth, D. E. (1998) The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, third edition, Addison-Wesley.
[81] Kontsevich, M. (2011) Noncommutative identities. arXiv:1109.2469v1
[82] Koutschan, C. (2008) Regular languages and their generating functions: The inverse problem. Theoret. Comput. Sci. 391 6574.
[83] Lagrange, J.-L. (1770) Nouvelle méthode pour résoudre les équations littérales par le moyen des séries. Histoire de l'Académie Royale des Sciences et des Belles Lettres de Berlin avec les Mémoires Tirez des Registres de Cette Académie XXIV 251326. Reprinted in Joseph Louis de Lagrange: Œuvres Complètes, Vol. 3, Gauthier-Villars (1869), pp. 5–73.
[84] Lalley, S. P. (1993) Finite range random walk on free groups and homogeneous trees. Ann. Probab. 21 571599.
[85] Lalley, S. P. (2004) Algebraic systems of generating functions and return probabilities for random walks. In Dynamics and Randomness II, Vol. 10 of Nonlinear Phenom. Complex Systems, Kluwer, pp. 81122.
[86] Lang, W. (2000) On generalizations of the Stirling number triangles. J. Integer Seq. 3 #00.2.4.
[87] Lothaire, M. (2005) Applied Combinatorics on Words, Vol. 105 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.
[88] Mansour, T. and Shattuck, M. (2011) Pattern avoiding partitions, sequence A054391 and the kernel method. Appl. Appl. Math. 6 397411.
[89] Meir, A. and Moon, J. W. (1978) On the altitude of nodes in random trees. Canad. J. Math. 30 9971015.
[90] Morcrette, B. (2012) Fully analyzing an algebraic Pólya urn model. In Proc. LATIN 2012, Vol. 7256 of Lecture Notes in Computer Science, Springer, pp. 568581.
[91] Morgenbesser, J. F. (2010) Square root singularities of infinite systems of functional equations. In 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms: AofA'10, DMTCS Proc. AM, pp. 513525.
[92] Muller, D. E. and Schupp, P. E. (1981) Context-free languages, groups, the theory of ends, second-order logic, tiling problems, cellular automata, and vector addition systems. Bull. Amer. Math. Soc. (NS) 4 331334.
[93] Pak, I. and Garrabrant, S. (2014) Counting with irrational tiles. Preprint.
[94] Panholzer, A. (2005) Gröbner bases and the defining polynomial of a context-free grammar generating function. J. Autom. Lang. Combin. 10 7997.
[95] Petre, I. and Salomaa, A. (2009) Algebraic systems and pushdown automata. Chapter 7 of Handbook of Weighted Automata (Droste, M. et al., eds), Monographs in Theoretical Computer Science, Springer, pp. 257289.
[96] Pevzner, P. A. and Waterman, M. S. (1995) Open combinatorial problems in computational molecular biology. In Third Israel Symposium on the Theory of Computing and Systems, IEEE Computer Society Press, pp. 158173.
[97] Pivoteau, C., Salvy, B. and Soria, M. (2012) Algorithms for combinatorial structures: Well-founded systems and Newton iterations. J. Combin. Theory Ser. A 119 17111773.
[98] Reutenauer, C. and Robado, M. (2012) On an algebraicity theorem of Kontsevich. In Proc. FPSAC '2012, DMTCS Proc. AR, pp. 239246.
[99] Richard, C. (2009) Limit distributions and scaling functions. Polygons, Polyominoes and Polycubes, Vol. 775 of Lecture Notes in Physics, Springer, pp. 247299.
[100] Rozenberg, G. and Salomaa, A., eds (1997) Handbook of Formal Languages (three volumes), Springer.
[101] Salomaa, A. (1990) Formal languages and power series. In Handbook of Theoretical Computer Science, Vol. B, Elsevier, pp. 103132.
[102] Salomaa, A. and Soittola, M. (1978) Automata-Theoretic Aspects of Formal Power Series, Texts and Monographs in Computer Science, Springer.
[103] Schneider, C. (2010) A symbolic summation approach to find optimal nested sum representations. In Motives, Quantum Field Theory, and Pseudodifferential Operators, Vol. 12 of Clay Mathematics Proceedings, AMS, pp. 285308.
[104] Schützenberger, M.-P. (1962) On a theorem of R. Jungen. Proc. Amer. Math. Soc. 13 885890.
[105] Schwarz, H. A. (1872) On these cases in which the Gaussian hypergeometric series represents an algebraic function of its fourth element. (Ueber diejenigen Fälle, in welchen die Gauss'sche hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt.) J. Reine Angewandte Mathematik LXXV 292335.
[106] Singer, M. F. (1980) Algebraic solutions of nth order linear differential equations. In Proc. Queen's Number Theory Conference 1979, Vol. 54 of Queen's Papers in Pure and Applied Mathematics, Queen's University, Kingston, Ontario, pp. 379420.
[107] Singer, M. F. (1986) Algebraic relations among solutions of linear differential equations. Trans. Amer. Math. Soc. 295 753763.
[108] Soittola, M. (1976) Positive rational sequences. Theoret. Comput. Sci. 2 317322.
[109] Sokal, A. D. (2009/10) A ridiculously simple and explicit implicit function theorem. Sém. Lothar. Combin. 61A #B61Ad, 21.
[110] Stanley, R. P. (1999) Enumerative Combinatorics 2, Vol. 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.
[111] Sturmfels, B. (2002) Solving Systems of Polynomial Equations, Vol. 97 of CBMS Regional Conference Series in Mathematics, Conference Board of the Mathematical Sciences.
[112] Tafat Bouzid, H. (2012) Combinatoire analytique des langages réguliers et algébriques. PhD thesis, Université Paris-XIII.
[113] Tannery, J. (1874) Propriétés des intégrales des équations différentielles linéaires à coefficients variables. Doctoral thesis, Faculté des Sciences de Paris.
[114] Trèves, F. (2006) Topological Vector Spaces, Distributions and Kernels, Dover. Unabridged republication of the 1967 original.
[115] van der Poorten, A. J. (1989) Some facts that should be better known, especially about rational functions. In Number Theory and Applications 1988, Vol. 265 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer, pp. 497528.
[116] van Leeuwen, J., ed. (1990) Handbook of Theoretical Computer Science, Vol. B: Formal Models and Semantics, Elsevier.
[117] Viennot, G. (1985) Enumerative combinatorics and algebraic languages. In Fundamentals of Computation Theory 1985, Vol. 199 of Lecture Notes in Computer Science, Springer, pp. 450464.
[118] Waterman, M. S. (1995) Applications of combinatorics to molecular biology. Chapter 39 of Handbook of Combinatorics, Vols 1, 2, Elsevier, pp. 19832001.
[119] Weinberg, F. and Nebel, M. E. (2010) Extending stochastic context-free grammars for an application in bioinformatics. In Language and Automata Theory and Applications, Vol. 6031 of Lecture Notes in Computer Science, Springer, pp. 585595.
[120] Woess, W. (2012) Context-free pairs of groups II: Cuts, tree sets, and random walks. Discrete Math. 312 157173.
[121] Woods, A. R. (1997) Coloring rules for finite trees, and probabilities of monadic second order sentences. Random Struct. Alg. 10 453485.



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