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Volumn 50, Issue 2, 2008, Pages 199-272

Twenty combinatorial examples of asymptotics derived from multivariate generating functions

Author keywords

Enumeration; Kernel method; Lagrange inversion; Multiple point; Residue; Singularity; Smooth point; Stratified Morse theory; Transfer matrix

Indexed keywords

ASYMPTOTIC ANALYSIS; GRAPH THEORY; MULTIVARIANT ANALYSIS; TRANSFER MATRIX METHOD;

EID: 44949173898     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/050643866     Document Type: Article
Times cited : (79)

References (72)
  • 1
    • 44949210263 scopus 로고    scopus 로고
    • I. A. AǏZENBERG AND A. P. YUZHAKOV, Integral Representations and Residues in Multidimensional Complex Analysis, Transl, Math. Monogr. 58, AMS, Providence, RI, 1983.
    • I. A. AǏZENBERG AND A. P. YUZHAKOV, Integral Representations and Residues in Multidimensional Complex Analysis, Transl, Math. Monogr. 58, AMS, Providence, RI, 1983.
  • 3
    • 0035539862 scopus 로고    scopus 로고
    • Random maps, coalescing saddles, singularity analysis, and Airy phenomena
    • C. BANDERIER, P. FLAJOLET, G. SCHAEFFER, AND M. SORIA, Random maps, coalescing saddles, singularity analysis, and Airy phenomena, Random Structures Algorithms, 19 (2001), pp. 194-246.
    • (2001) Random Structures Algorithms , vol.19 , pp. 194-246
    • BANDERIER, C.1    FLAJOLET, P.2    SCHAEFFER, G.3    SORIA, M.4
  • 4
    • 44949101133 scopus 로고    scopus 로고
    • Generating functioning with poles on a hyperplane arrangement
    • in preparation
    • Y. BARYSHNIKOV AND R. PEMANTLE, Generating functioning with poles on a hyperplane arrangement, in preparation.
    • BARYSHNIKOV, Y.1    PEMANTLE, R.2
  • 5
    • 0015896876 scopus 로고
    • Central and local limit theorems applied to asymptotic enumeration
    • E. A. BENDER, Central and local limit theorems applied to asymptotic enumeration, J. Combin. Theory Ser. A, 15 (1973), pp. 91-111.
    • (1973) J. Combin. Theory Ser. A , vol.15 , pp. 91-111
    • BENDER, E.A.1
  • 6
    • 0016114863 scopus 로고
    • Asymptotic methods in enumeration
    • E. A. BENDER, Asymptotic methods in enumeration, SIAM Rev., 16 (1974), pp. 485-515.
    • (1974) SIAM Rev , vol.16 , pp. 485-515
    • BENDER, E.A.1
  • 8
    • 0008398347 scopus 로고    scopus 로고
    • E. A. BENDER AND L. B. RICHMOND, Multivariate asymptotics for products of large powers with applications to Lagrange inversion, Electron. J. Combin., 6 (1999), article 8 (electronic).
    • E. A. BENDER AND L. B. RICHMOND, Multivariate asymptotics for products of large powers with applications to Lagrange inversion, Electron. J. Combin., 6 (1999), article 8 (electronic).
  • 10
    • 0027610655 scopus 로고
    • Multidimensional residues, generating functions, and their application to queueing networks
    • A. BERTOZZI AND J. MCKENNA, Multidimensional residues, generating functions, and their application to queueing networks, SIAM Rev., 35 (1993), pp. 239-268.
    • (1993) SIAM Rev , vol.35 , pp. 239-268
    • BERTOZZI, A.1    MCKENNA, J.2
  • 13
    • 0001937215 scopus 로고    scopus 로고
    • Local statistics for random domino tilings of the Aztec diamond
    • H. COHN, N. ELKIES, AND J. PROPP, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J., 85 (1996), pp. 117-166.
    • (1996) Duke Math. J , vol.85 , pp. 117-166
    • COHN, H.1    ELKIES, N.2    PROPP, J.3
  • 14
    • 0004258018 scopus 로고
    • D. Reidel Publishing, Dordrecht, The Netherlands
    • L. COMTET, Advanced Combinatorics, D. Reidel Publishing, Dordrecht, The Netherlands, 1974.
    • (1974) Advanced Combinatorics
    • COMTET, L.1
  • 16
    • 3242656629 scopus 로고    scopus 로고
    • Algebraic unimodular counting
    • J. A. DE LOERA AND B. STURMFELS, Algebraic unimodular counting, Math. Program., 96 (2003), pp. 183-203.
    • (2003) Math. Program , vol.96 , pp. 183-203
    • DE LOERA, J.A.1    STURMFELS, B.2
  • 17
    • 0035965434 scopus 로고    scopus 로고
    • A survey of the Fine numbers
    • E. DEUTSCH AND L. SHAPIRO, A survey of the Fine numbers, Discrete Math., 241 (2001), pp. 241-265.
    • (2001) Discrete Math , vol.241 , pp. 241-265
    • DEUTSCH, E.1    SHAPIRO, L.2
  • 18
    • 38149147155 scopus 로고
    • A bivariate asymptotic expansion of coefficients of powers of generating functions
    • M. DRMOTA, A bivariate asymptotic expansion of coefficients of powers of generating functions, European J. Combin., 15 (1994), pp. 139-152.
    • (1994) European J. Combin , vol.15 , pp. 139-152
    • DRMOTA, M.1
  • 20
    • 0000640693 scopus 로고
    • Singularity analysis of generating functions
    • P. FLAJOLET AND A. ODLYZKO, Singularity analysis of generating functions, SIAM J. Discrete Math., 3 (1990), pp. 216-240.
    • (1990) SIAM J. Discrete Math , vol.3 , pp. 216-240
    • FLAJOLET, P.1    ODLYZKO, A.2
  • 21
    • 44949245718 scopus 로고    scopus 로고
    • P. FLAJOLET AND R. SEDGEWICK, Analytic combinatorics, in preparation. Version 0.7 (2006) downloaded from
    • P. FLAJOLET AND R. SEDGEWICK, Analytic combinatorics, in preparation. Version 0.7 (2006) downloaded from http://algo.inria.fr/ flajolet/Publications/books.html.
  • 22
    • 3042529257 scopus 로고    scopus 로고
    • A. FLAXMAN, A. W. HARROW, AND G. B. SORKIN, Strings with maximally many distinct subsequences and substrings, Electron. J. Combin., 11 (2004), article 8 (electronic).
    • A. FLAXMAN, A. W. HARROW, AND G. B. SORKIN, Strings with maximally many distinct subsequences and substrings, Electron. J. Combin., 11 (2004), article 8 (electronic).
  • 23
    • 0000543521 scopus 로고
    • Algebraic functions over finite fields
    • H. FURSTENBERG, Algebraic functions over finite fields, J. Algebra, 7 (1967), pp. 271-277.
    • (1967) J. Algebra , vol.7 , pp. 271-277
    • FURSTENBERG, H.1
  • 24
    • 38049026666 scopus 로고    scopus 로고
    • Z. GAO AND L. B. RICHMOND, Central and local limit theorems applied to asymptotic enumeration. IV. Multivariate generating functions, J. Comput. Appl. Math., 41 (1992), pp. 177-186.
    • Z. GAO AND L. B. RICHMOND, Central and local limit theorems applied to asymptotic enumeration. IV. Multivariate generating functions, J. Comput. Appl. Math., 41 (1992), pp. 177-186.
  • 25
    • 0032661468 scopus 로고    scopus 로고
    • Z. GAO AND N. C. WORMALD, The size of the largest components in random planar maps, SIAM J. Discrete Math., 12 (1999), pp. 217-228.
    • Z. GAO AND N. C. WORMALD, The size of the largest components in random planar maps, SIAM J. Discrete Math., 12 (1999), pp. 217-228.
  • 26
    • 0347164911 scopus 로고    scopus 로고
    • Z. GAO AND N. C. WORMALD, The distribution of the maximum vertex degree in random planar maps, J. Combin. Theory Ser. A, 89 (2000), pp. 201-230.
    • Z. GAO AND N. C. WORMALD, The distribution of the maximum vertex degree in random planar maps, J. Combin. Theory Ser. A, 89 (2000), pp. 201-230.
  • 27
    • 0007468685 scopus 로고
    • Some results on the asymptotic behaviour of coefficients of large powers of functions
    • D. GARDY, Some results on the asymptotic behaviour of coefficients of large powers of functions, Discrete Math., 139 (1995), pp. 189-217.
    • (1995) Discrete Math , vol.139 , pp. 189-217
    • GARDY, D.1
  • 28
    • 0003353305 scopus 로고
    • Discriminants, Resultants, and Multidimensional Determinants, Math
    • Birkhäuser Boston, Boston, MA
    • I. M. GEL'FAND, M. M. KAPRANOV, AND A. V. ZELEVINSKY, Discriminants, Resultants, and Multidimensional Determinants, Math. Theory Appl., Birkhäuser Boston, Boston, MA, 1994.
    • (1994) Theory Appl
    • GEL'FAND, I.M.1    KAPRANOV, M.M.2    ZELEVINSKY, A.V.3
  • 29
    • 38249009516 scopus 로고
    • Super ballot numbers
    • I. M. GESSEL, Super ballot numbers, J. Symbolic Comput., 14 (1992), pp. 179-194.
    • (1992) J. Symbolic Comput , vol.14 , pp. 179-194
    • GESSEL, I.M.1
  • 30
    • 43449116273 scopus 로고
    • On elementary methods in positivity theory
    • J. GILLIS, B. REZNICK, AND D. ZEILBERGER, On elementary methods in positivity theory, SIAM J. Math. Anal., 14 (1983), pp. 396-398.
    • (1983) SIAM J. Math. Anal , vol.14 , pp. 396-398
    • GILLIS, J.1    REZNICK, B.2    ZEILBERGER, D.3
  • 35
    • 34250201608 scopus 로고
    • String overlaps, pattern matching, and nontransitive games
    • L. J. GUIBAS AND A. M. ODLYZKO, String overlaps, pattern matching, and nontransitive games, J. Combin. Theory Ser. A, 30 (1981), pp. 183-208.
    • (1981) J. Combin. Theory Ser. A , vol.30 , pp. 183-208
    • GUIBAS, L.J.1    ODLYZKO, A.M.2
  • 36
    • 84960557830 scopus 로고
    • Asymptotic formulae for the distribution of integers of various types
    • G. HARDY AND S. RAMANUJAN, Asymptotic formulae for the distribution of integers of various types, Proc. London Math. Soc., 16 (1917), pp. 112-132.
    • (1917) Proc. London Math. Soc , vol.16 , pp. 112-132
    • HARDY, G.1    RAMANUJAN, S.2
  • 37
    • 84972547740 scopus 로고
    • The diagonal of a double power series
    • M. L. J. HAUTUS AND D. A. KLARNER, The diagonal of a double power series, Duke Math. J., 38 (1971), pp. 229-235.
    • (1971) Duke Math. J , vol.38 , pp. 229-235
    • HAUTUS, M.L.J.1    KLARNER, D.A.2
  • 38
    • 0002121590 scopus 로고
    • A generalisation of Stirling's formula
    • W. K. HAYMAN, A generalisation of Stirling's formula, J. Reine Angew. Math., 196 (1956), pp. 67-95.
    • (1956) J. Reine Angew. Math , vol.196 , pp. 67-95
    • HAYMAN, W.K.1
  • 43
    • 0036771519 scopus 로고    scopus 로고
    • Asymptotic expansions for large closed and loss queueing networks
    • Y. KOCAN, Asymptotic expansions for large closed and loss queueing networks, Math. Probl. Eng., 8 (2002), pp. 323-348.
    • (2002) Math. Probl. Eng , vol.8 , pp. 323-348
    • KOCAN, Y.1
  • 44
    • 0033444398 scopus 로고    scopus 로고
    • Coalescing particles on an interval
    • M. LARSEN AND R. LYONS, Coalescing particles on an interval, J. Theoret. Probab., 12 (1999), pp. 201-205.
    • (1999) J. Theoret. Probab , vol.12 , pp. 201-205
    • LARSEN, M.1    LYONS, R.2
  • 45
    • 0002974378 scopus 로고
    • Le calcul différentiel et intégral sur une variété analytique complexe, (Problème de Cauchy
    • J. LERAY, Le calcul différentiel et intégral sur une variété analytique complexe, (Problème de Cauchy. III), Bull. Soc. Math. France, 87 (1959), pp. 81-180.
    • (1959) III), Bull. Soc. Math. France , vol.87 , pp. 81-180
    • LERAY, J.1
  • 46
    • 0001530766 scopus 로고
    • A martingale approach to the study of occurrence of sequence patterns in repeated experiments
    • S.-Y. R. LI, A martingale approach to the study of occurrence of sequence patterns in repeated experiments, Ann. Probab., 8 (1980), pp. 1171-1176.
    • (1980) Ann. Probab , vol.8 , pp. 1171-1176
    • LI, S.-Y.R.1
  • 47
    • 84974004647 scopus 로고
    • The asymptotics of a lattice point problem associated to a finite number of polynomials
    • B. LICHTIN, The asymptotics of a lattice point problem associated to a finite number of polynomials. I, Duke Math. J., 63 (1991), pp. 139-192.
    • (1991) I, Duke Math. J , vol.63 , pp. 139-192
    • LICHTIN, B.1
  • 48
    • 38249027261 scopus 로고
    • The diagonal of a D-finite power series is D-finite
    • L. LIPSHITZ, The diagonal of a D-finite power series is D-finite, J. Algebra, 113 (1988), pp. 373-378.
    • (1988) J. Algebra , vol.113 , pp. 373-378
    • LIPSHITZ, L.1
  • 49
    • 36249019207 scopus 로고    scopus 로고
    • Uniform formulae for coefficients of meromorphic functions in two variables. Part I
    • M. LLADSER, Uniform formulae for coefficients of meromorphic functions in two variables. Part I, SIAM J. Discrete Math., 20 (2006), pp. 811-828.
    • (2006) SIAM J. Discrete Math , vol.20 , pp. 811-828
    • LLADSER, M.1
  • 52
    • 0010831641 scopus 로고    scopus 로고
    • A. MEIR AND J. W. MOON, The asymptotic behaviour of coefficients of powers of certain generating functions, European J. Combin., 11 (1990), pp. 581-587.
    • A. MEIR AND J. W. MOON, The asymptotic behaviour of coefficients of powers of certain generating functions, European J. Combin., 11 (1990), pp. 581-587.
  • 53
    • 0003521409 scopus 로고
    • Algebraic Geometry
    • Springer-Verlag, Berlin
    • D. MUMFORD, Algebraic Geometry. I, Classics Math., Springer-Verlag, Berlin, 1995.
    • (1995) I, Classics Math
    • MUMFORD, D.1
  • 54
    • 44949243849 scopus 로고    scopus 로고
    • Sém. Lothar. Combin, 49 (2002/2004, article B49h electronic
    • E. MUNARINI AND N. ZAGAGLIA SALVI, Binary strings without zigzags, Sém. Lothar. Combin., 49 (2002/2004), article B49h (electronic); http://www.mat.univie.at/nsic/wpapers/s49/html.
    • Binary strings without zigzags
    • MUNARINI, E.1    ZAGAGLIA SALVI, N.2
  • 55
    • 0000278117 scopus 로고
    • Asymptotic enumeration methods
    • Vols, Elsevier, Amsterdam, The Netherlands
    • A. M. ODLYZKO, Asymptotic enumeration methods, in Handbook of Combinatorics, Vols. 1, 2, Elsevier, Amsterdam, The Netherlands, 1995, pp. 1063-1229.
    • (1995) Handbook of Combinatorics , vol.1-2 , pp. 1063-1229
    • ODLYZKO, A.M.1
  • 59
    • 20344385946 scopus 로고    scopus 로고
    • An arctic circle theorem for groves
    • T. K. PETERSEN AND D. SPEYER, An arctic circle theorem for groves, J. Combin. Theory Ser. A, 111 (2005), pp. 137-164.
    • (2005) J. Combin. Theory Ser. A , vol.111 , pp. 137-164
    • PETERSEN, T.K.1    SPEYER, D.2
  • 60
    • 33646983172 scopus 로고
    • On the number of certain lattice polygons
    • G. PÓLYA, On the number of certain lattice polygons, J. Combin. Theory, 6 (1969), pp. 102-105.
    • (1969) J. Combin. Theory , vol.6 , pp. 102-105
    • PÓLYA, G.1
  • 61
    • 44949236888 scopus 로고    scopus 로고
    • Generating functions in computational biology, J. Iranian Statist. Soc
    • to appear
    • M. RÉGNIER AND F. TAHI, Generating functions in computational biology, J. Iranian Statist. Soc., to appear.
    • RÉGNIER, M.1    TAHI, F.2
  • 63
    • 0039895899 scopus 로고
    • Pascal triangles, Catalan numbers and renewal arrays
    • D. G. ROGERS, Pascal triangles, Catalan numbers and renewal arrays, Discrete Math., 22 (1978), pp. 301-310.
    • (1978) Discrete Math , vol.22 , pp. 301-310
    • ROGERS, D.G.1
  • 65
    • 0001731038 scopus 로고
    • Riordan arrays and combinatorial sums
    • R. SPRUGNOLI, Riordan arrays and combinatorial sums, Discrete Math., 132 (1994), pp. 267-290.
    • (1994) Discrete Math , vol.132 , pp. 267-290
    • SPRUGNOLI, R.1
  • 66
    • 0003526491 scopus 로고    scopus 로고
    • 1, Cambridge Stud. Adv. Math. 49, Cambridge University Press, Cambridge, UK
    • R. P. STANLEY, Enumerative Combinatorics. Vol. 1, Cambridge Stud. Adv. Math. 49, Cambridge University Press, Cambridge, UK, 1997.
    • (1997) Enumerative Combinatorics. Vol
    • STANLEY, R.P.1
  • 67
    • 0003652674 scopus 로고    scopus 로고
    • Cambridge Stud. Adv. Math. 62, Cambridge University Press, Cambridge, UK
    • R. P. STANLEY, Enumerative Combinatorics. Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge University Press, Cambridge, UK, 1999.
    • (1999) Enumerative Combinatorics , vol.2
    • STANLEY, R.P.1
  • 68
    • 33645088458 scopus 로고    scopus 로고
    • What is... a Gröbner basis?
    • B. STURMFELS, What is... a Gröbner basis?, Notices Amer. Math. Soc., 52 (2005), pp. 1199-1200.
    • (2005) Notices Amer. Math. Soc , vol.52 , pp. 1199-1200
    • STURMFELS, B.1
  • 69
    • 44949265123 scopus 로고    scopus 로고
    • Automatic computation of asymptotics of coefficients of bivariate rational generating functions
    • in preparation
    • J. VAN DER HOEVEN AND R. PEMANTLE, Automatic computation of asymptotics of coefficients of bivariate rational generating functions, in preparation.
    • VAN DER HOEVEN, J.1    PEMANTLE, R.2
  • 70
    • 0041938733 scopus 로고
    • Applications of combinatorics to molecular biology
    • Vols, Elsevier, Amsterdam, The Netherlands
    • M. S. WATERMAN, Applications of combinatorics to molecular biology, in Handbook of Combinatorics, Vols. 1, 2, Elsevier, Amsterdam, The Netherlands, 1995, pp. 1983-2001.
    • (1995) Handbook of Combinatorics , vol.1-2 , pp. 1983-2001
    • WATERMAN, M.S.1
  • 72
    • 44949107138 scopus 로고    scopus 로고
    • M.C. WILSON, Asymptotics for generalized Riordan arrays, in Proceedings of the 2005 International Conference on Analysis of Algorithms, Discrete Math. Theor. Comput. Sci. Proc., AD, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2005, pp. 323-333 (electronic).
    • M.C. WILSON, Asymptotics for generalized Riordan arrays, in Proceedings of the 2005 International Conference on Analysis of Algorithms, Discrete Math. Theor. Comput. Sci. Proc., AD, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2005, pp. 323-333 (electronic).


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