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  • Cited by 182
Publisher:
Cambridge University Press
Online publication date:
April 2015
Print publication year:
2015
Online ISBN:
9781139871495

Book description

Catalan numbers are probably the most ubiquitous sequence of numbers in mathematics. This book gives for the first time a comprehensive collection of their properties and applications to combinatorics, algebra, analysis, number theory, probability theory, geometry, topology, and other areas. Following an introduction to the basic properties of Catalan numbers, the book presents 214 different kinds of objects counted by them in the form of exercises with solutions. The reader can try solving the exercises or simply browse through them. Some 68 additional exercises with prescribed difficulty levels present various properties of Catalan numbers and related numbers, such as Fuss-Catalan numbers, Motzkin numbers, Schröder numbers, Narayana numbers, super Catalan numbers, q-Catalan numbers and (q,t)-Catalan numbers. The book ends with a history of Catalan numbers by Igor Pak and a glossary of key terms. Whether your interest in mathematics is recreation or research, you will find plenty of fascinating and stimulating facts here.

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Contents

Bibliography
[1] C., Aebi and G., Cairns, Catalan numbers, primes, and twin primes, Elem. Math. 63 (2008 Google Scholar), 153–164.
[2] M., Aigner, Catalan and other numbers, in Algebraic Combinatorics and Computer Science, Springer, Berlin, 2001 Google Scholar, 347–390.
[3] D., André, Solution directe du problème résolu par M. Bertrand, Comptes Rendus Acad. Sci. Paris 105 (1887 Google Scholar), 436–437.
[4] G. E., Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, 1976 Google Scholar.
[5] É., Barbier, Généralisation du problème résolu par M.J. Bertrand, Comptes Rendus Acad. Sci. Paris 105 (1887 Google Scholar), 407.
[6] E. T., Bell, The iterated exponential integers, Annals of Math. 39 (1938 Google Scholar), 539–557.
[7] J., Bertrand, Solution d'une problème, Comptes Rendus Acad. Sci. Paris 105 (1887 Google Scholar), 369.
[8] M.J., Binet, Réflexions sur le problème dédeterminer le nombre de manières dont une figure rectiligne peut être partagées en triangles au moyen de ses diagonales, J. Math. Pures Appl. 4 (1839 Google Scholar), 79–90.
[9] W. G., Brown, Historical note on a recurrent combinatorial problem, Amer. Math. Monthly 72 (1965 Google Scholar), 973–979.
[10] B., Bru, Les leçons de calcul des probabilités de Joseph Bertrand, J. Électron. Hist. Probab. Stat. 2 (2006 Google Scholar), no. 2, 44 pp.
[11] N. G., de Bruijn and B. J. M., Morselt, A note on plane trees, J. Combinatorial Theory 2 (1967 Google Scholar), 27–34.
[12] R. S., Calinger, Leonhard Euler: life and thought, in Leonhard Euler: Life, Work and Legacy (R. E., Bradley and E., Sandifer, editors), Elsevier, Amsterdam, 2007 Google Scholar.
[13] P. J., Cameron Google Scholar, LTCC course notes on Enumerative Combinatorics, Lecture 3: Catalan numbers (Autumn 2013); available at http://www.maths.qmul.ac.uk/~pjc/ec/.
[14] E. C., Catalan, Note sur une équation aux différences finies, J. Math. pure et appliquées 3 (1838 Google Scholar), 508–516.
[15] E. C., Catalan, Solution nouvelle de cette question: un polygone étant donné, de combien de manières peut-on le partager en triangles au moyen de diagonales?, J. Math. Pures Appl. 4 (1839 Google Scholar), 91–94.
[16] E., Catalan, Sur les nombres de Segner, Rend. Circ. Mat. Palermo 1 (1887 Google Scholar), 190–201.
[17] A., Cayley, On the analytical forms called trees II, Philos. Mag. 18 (1859 Google Scholar), 374–378.
[18] A., Cayley, On the partitions of a polygon, Proc. London Math. Soc. 22 (1891 Google Scholar), 237–262.
[19] A., Dvoretzky and Th., Motzkin, A problem of arrangements, Duke Math. J. 14 (1947 Google Scholar), 305–313.
[20] L., Euler Google Scholar, Summary of [61] in the same volume of Novi Commentarii, 13–15.
[21] E. A., Fellmann, Leonhard Euler, Birkhäuser, Basel, 2007 Google Scholar.
[22] N., Fuss, Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat, Novi Acta Acad. Sci. Petrop. 9 (1795 Google Scholar), 243–251; available at [55].
[23] M., Gardner, Mathematical Games, Catalan numbers: an integer sequence that materializes in unexpected places, Scientific Amer. 234, no. 6 (June 1976 Google Scholar), 120–125, 132.
[24] I. M., Gelfand, M. M., Kapranov, and A. V., Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser Boston, Cambridge, Massachusetts, 1994 Google Scholar.
[25] H. W., Gould Google Scholar, Bell and Catalan Numbers: A Research Bibliography of Two Special Number Sequences, Revised 2007 edition; available at http://tinyurl.com/opknlh8.
[26] I. P., Goulden and D. M., Jackson, Combinatorial Enumeration, John Wiley, New York, 1983 Google Scholar; reissued by Dover, New York, 2004.
[27] J.A.S., Growney, Finitely generated free groupoids, PhD thesis, University of Oklahoma, 1970 Google Scholar.
[28] J. A., Grunert, Ueber die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein neck durch Diagonales in lauter mecke zerlegen lässt, mit Bezug auf einige Abhandlungen der Herren Lamé, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathématiques Pure et Appliquées, publié par Joseph Liouville, Vols. 3, 4, Arch. Math. Physik 1 (1841 Google Scholar), 193–203.
[29] M., Hall, Combinatorial Theory, Blaisdell, Waltham, Massachusetts, 1967 Google Scholar.
[30] K., Humphreys, A history and a survey of lattice path enumeration, J. Statist. Plann. Inference 140 (2010 Google Scholar), 2237–2254.
[31] M., Kauers and P., Paule, The Concrete Tetrahedron, Springer, Vienna, 2011 Google Scholar.
[32] D. A., Klarner, Correspondences between plane trees and binary sequences, J. Combinatorial Theory 9 (1970 Google Scholar), 401–411.
[33] D. E., Knuth, The Art of Computer Programming, vol. 1, Fundamental Algorithms, Addison-Wesley, Reading, Massachusetts, 1968 Google Scholar; second ed., 1973.
[34] D. E., Knuth, The Art of Computer Programming, vol. 3, 2nd ed., Addison-Wesley, Reading, Massachusetts, 1998 Google Scholar.
[35] T., Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009 Google Scholar.
[36] H. S., Kim, Interview with Professor Stanley, Math Majors Magazine, vol. 1, no. 1 (December 2008 Google Scholar), pp. 28–33; available at http://tinyurl.com/q423c6l.
[37] T. P., Kirkman, On the K-Partitions of the R-Gon and R-Ace, Phil. Trans. Royal Soc. 147 (1857 Google Scholar), 217–272.
[38] S., Kotelnikow, Demonstatio seriei exhibitae in recensione VI. tomi VII. Commentariorum A. S. P., Novi Comment. Acad. Sci. Imp. Petropol. 10 (1766 Google Scholar), 199–204; available at [55].
[39] G., Lamé, Extrait d'une lettre de M. Lamé à M. Liouville sur cette question: Un polygone convexe étant donné, de combien de manières peut-on le partager en triangles au moyen de diagonales?, Journal de Mathématiques pure et appliquées 3 (1838 Google Scholar), 505–507.
[40] P. J., Larcombe, The 18th century Chinese discovery of the Catalan numbers, Math. Spectrum 32 (1999 Google Scholar/2000), 5–7.
[41] P. J., Larcombe, On pre-Catalan Catalan numbers: Kotelnikow (1766), Math. Today 35 (1999 Google Scholar), no. 1, 25.
[42] P. J., Larcombe and P.D.C., Wilson, On the trail of the Catalan sequence, Mathematics Today 34 (1998 Google Scholar), 114–117.
[43] P. J., Larcombe and P.D.C., Wilson, On the generating function of the Catalan sequence, Congr. Numer. 149 (2001 Google Scholar), 97–108.
[44] J., Liouville, Remarques sur un mémoire de N. Fuss, J. Math. Pures Appl. 8 (1843 Google Scholar), 391–394.
[45] E., Lucas, Théorie des Nombres, Gauthier-Villard, Paris, 1891 Google Scholar.
[46] J. J., Luo, Antu Ming, the first inventor of Catalan numbers in the world, Neimengu Daxue Xuebao 19 (1988 Google Scholar), 239–245 (in Chinese).
[47] J., Luo, Ming Antu and his power series expansions, in Seki, founder of modern mathematics in Japan, Springer, Tokyo, 2013 Google Scholar, 299–310.
[48] P. A., MacMahon, Combinatory Analysis, vols. 1 and 2, Cambridge University Press, 1916 Google Scholar; reprinted by Chelsea, New York, 1960, and by Dover, New York, 2004.
[49] J., McCammond, Noncrossing partitions in surprising locations, Amer. Math. Monthly 113 (2006 Google Scholar), 598–610.
[50] D., Mirimanoff, A propos de l'interprétation géométrique du problème du scrutin, L'Enseignement Math. 23 (1923 Google Scholar), 187–189.
[51] S. G., Mohanty, Lattice Path Counting and Applications, Academic Press, New York, 1979 Google Scholar.
[52] T. V., Narayana, Lattice Path Combinatorics with Statistical Applications, Mathe-matical Expositions no. 23, University of Toronto Press, Toronto, 1979 Google Scholar.
[53] E., Netto, Lehrbuch der Combinatorik, Teubner, Leipzig, 1901 Google Scholar.
[54] I., Pak Google Scholar, Who computed Catalan numbers? (February 20, 2013), Who named Catalan numbers? (February 5, 2014), blog posts on Igor Pak's blog; available at http://igorpak.wordpress.com/.
[55] I., Pak Google Scholar, Catalan Numbers website, http://www.math.ucla.edu/~pak/lectures/Cat/pakcat.htm.
[56] M., Renault, Lost (and found) in translation, Amer. Math. Monthly 115 (2008 Google Scholar), 358–363.
[57] J., Riordan, Combinatorial Identities, John Wiley, New York, 1968 Google Scholar.
[58] O., Rodrigues, Sur le nombre de manières de décomposer un polygone en triangles au moyen de diagonales, J. Math. Pures Appl. 3 (1838 Google Scholar), 547–548.
[59] O., Rodrigues, Sur le nombre de manières d'effectuer un produit de n facteurs, J. Math. Pures Appl. 3 (1838 Google Scholar), 549.
[60] E., Schröder, Vier combinatorische Probleme, Z. für Math. Phys. 15 (1870 Google Scholar), 361–376.
[61] J. A., Segner, Enumeratio modorum quibus figurae planae rectilinae per diagonales dividuntur in triangula, Novi Comment. Acad. Sci. Imp. Petropol. 7 (dated 1758 Google Scholar/59, published in 1761), 203–210; available at [55].
[62] N.J.A., Sloane, A Handbook of Integer Sequences, Academic Press, New York, 1973 Google Scholar.
[63] N.J.A., Sloane and S., Plouffe, The Encyclopedia of Integer Sequences, Academic Press, New York, 1995 Google Scholar.
[64] R., Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Cambridge University Press, Cambridge, 2012 Google Scholar.
[65] R., Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, New York/Cambridge, 1999 Google Scholar.
[66] L., Takács, On the ballot theorems, in Advances in Combinatorial Methods and Applications to Probability and Statistics, Birkhäuser, Boston, Massachusetts, 1997 Google Scholar, 97–114.
[67] U., Tamm, Olinde Rodrigues and combinatorics, in Mathematics and Social Utopias in France, Olinde Rodrigues and His Times (S., Altmann and E. L., Ortiz, eds.), American Mathematical Society and London Mathematical Society, Providence Google Scholar, Rhode Island, pp. 119–129.
[68] H. M., Taylor and R. C., Rowe, Note on a geometrical theorem, Proc. London Math. Soc. 13 (1882 Google Scholar), 102–106.
[69] H.N.V., Temperley and E. H., Lieb, Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem, Proc. Royal Soc. London, Ser. A 322 (1971 Google Scholar), 251–280.
[70] J. H., van Lint, Combinatorial Theory Seminar, Lecture Notes in Math. 382, Springer, Berlin, 1974 Google Scholar, 131 pp.
[71] V. S., Varadarajan, Euler Through Time: A New Look at Old Themes, AMS, Providence, Rhode Island, 2006 Google Scholar.
[72] W. A., Whitworth, Arrangements of m things of one sort and n things of another sort, under certain conditions of priority, Messenger of Math. 8 (1879 Google Scholar), 105–114.
[73] B., Ycart, A case of mathematical eponymy: the Vandermonde determinant, Rev. Histoire Math. 19 (2013 Google Scholar), 43–77.
[74] D., Zeilberger Google Scholar, Opinion 49 (Oct. 25, 2002); available at http://tinyurl.com/pzp8bvk.

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