## Catalan Numbers with ApplicationsLike the intriguing Fibonacci and Lucas numbers, Catalan numbers are also ubiquitous. "They have the same delightful propensity for popping up unexpectedly, particularly in combinatorial problems," Martin Gardner wrote in Scientific American. "Indeed, the Catalan sequence is probably the most frequently encountered sequence that is still obscure enough to cause mathematicians lacking access to Sloane's Handbook of Integer Sequences to expend inordinate amounts of energy re-discovering formulas that were worked out long ago," he continued. As Gardner noted, many mathematicians may know the abc's of Catalan sequence, but not many are familiar with the myriad of their unexpected occurrences, applications, and properties; they crop up in chess boards, computer programming, and even train tracks. This book presents a clear and comprehensive introduction to one of the truly fascinating topics in mathematics. Catalan numbers are named after the Belgian mathematician Eugene Charles Catalan (1814-1894), who "discovered" them in 1838, though he was not the first person to discover them. The great Swiss mathematician Leonhard Euler (1707-1763) "discovered" them around 1756, but even before then and though his work was not known to the outside world, Chinese mathematician Antu Ming (1692?-1763) first discovered Catalan numbers about 1730. Catalan numbers can be used by teachers and professors to generate excitement among students for exploration and intellectual curiosity and to sharpen a variety of mathematical skills and tools, such as pattern recognition, conjecturing, proof-techniques, and problem-solving techniques. This book is not only intended for mathematicians but for a much larger audience, including high school students, math and science teachers, computer scientists, and those amateurs with a modicum of mathematical curiosity. An invaluable resource book, it contains an intriguing array of applications to computer science, abstract algebra, combinatorics, geometry, graph theory, chess, and World Series. |

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### Contents

1 | |

2 The Central Binomial Coefficient | 13 |

3 The Central Binomial Coefficient Revisited | 51 |

4 Binomial Coefficients Revisited | 89 |

5 Catalan Numbers | 103 |

6 The Ubiquity of Catalan Numbers I | 149 |

7 The Ubiquity of Catalan Numbers II | 191 |

8 Trees and Catalan Numbers | 227 |

13 Divisibility Properties | 329 |

14 A Catalan Triangle | 333 |

15 A Family of Binary Words | 347 |

16 Tribinomial Coefficients | 357 |

17 Generalized Catalan Numbers | 375 |

Appendix A | 379 |

The First 100 Catalan Numbers | 391 |

395 | |

9 Lattice Paths and Catalan Numbers | 259 |

10 Partitions and Catalan Numbers | 281 |

11 Algebra Sports and Catalan Numbers | 289 |

12 Pascals Triangle and Catalan Numbers | 313 |

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algebra algorithm bijection binary words binomial coefficients binomial theorem cartesian plane Catalan numbers Central Binomial Coefficient compute consider Corollary denote the number desired property downstep Dyck paths edges elements Erdös exactly explicit formula Fibonacci Find the number full binary trees function graph induction integer Interestingly label lattice paths lattice point Lemma line segment mathematician mathematics matrices mountain ranges multiset n-gon n-tuples n-tuples a 1a2 noncrossing partitions Notice number of paths number theory occurrences of Catalan ordered rooted trees pairs partial sum Pascal’s triangle permutations planted trivalent binary positive integers postfix Proof q q q recurrence relation recursive recursive formula resulting reversible set of binary shows the various Solution Figure Solution Table stack substring Suppose total number tree in Figure triangular array tribinomial trivalent binary tree University valid postfix expression various possible vertex vertices well-formed sequences yields