By Steven Roman

ISBN-10: 3319221434

ISBN-13: 9783319221434

This textbook presents an creation to the Catalan numbers and their extraordinary homes, in addition to their numerous purposes in combinatorics. Intended to be obtainable to scholars new to the topic, the publication starts with extra straightforward themes sooner than progressing to extra mathematically refined topics. Each bankruptcy specializes in a particular combinatorial item counted via those numbers, together with paths, timber, tilings of a staircase, null sums in Zn+1, period constructions, walls, diversifications, semiorders, and more. Exercises are integrated on the finish of ebook, besides tricks and ideas, to aid scholars receive a greater take hold of of the material. The textual content is perfect for undergraduate scholars learning combinatorics, yet also will attract an individual with a mathematical heritage who has an curiosity in studying concerning the Catalan numbers.

“Roman does an admirable activity of supplying an advent to Catalan numbers of a special nature from the former ones. He has made a very good number of subject matters in an effort to express the flavour of Catalan combinatorics. [Readers] will gather an excellent feeling for why such a lot of mathematicians are enthralled through the awesome ubiquity and style of Catalan numbers.”

 - From the foreword through Richard Stanley

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Read e-book online An Introduction to Catalan Numbers PDF

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This defines a family of injective maps θn, k : S ½k1;nŠ ! S ½1, kÀ1Š  S ½kþ1, nÀ1Š by θn, k ðF 1 [ F 2 [ f½k; nŠgÞ ¼ ðF 1 ; F 2 Þ Moreover, since any pair ðF 1 ; F 2 Þ 2 S ½1, kÀ1Š  S ½kþ1, nÀ1Š can be combined into a family F 1 [ F 2 [ f½k; nŠg in S ½k1;nŠ the map θn,k is a bijection. Set DÀ1 ¼ D0 ¼ 1. Also, D1 ¼ 2 since both the empty family and the family {[1]} are separated. 1 Cn counts the number of separated families of intervals in Intð½n À 1ŠÞ. □ Covering Antichains in Int([n]) The Catalan numbers count the number of covering antichains in the interval poset Int ([n]), that is, antichains in Int([n]) with the property that every element of [n] is contained in some interval of the antichain.

1 Cn counts the number of ordered trees with n þ 1 vertices. □ Binary Trees Let Bn be the family of binary trees with n vertices for n ! 1. 2. The roots for the subtrees are the vertices adjacent to the original root. 2 Decomposition of a binary tree If Bn,k is the family of all members of Bn that have a left subtree of size k, for 0 k n À 1, then this decomposition into a left and right subtrees defines a family of bijections θn, k : Bn, k ! Bk  BnÀkÀ1 by θn, k ðT Þ ¼ ðT ‘ ; T r Þ Hence, the number Dn of binary trees with n vertices satisfies the recurrence Dn ¼ nÀ1 X Dk DnÀkÀ1 k¼0 for n !

A: □ 4 Catalan Numbers and Paths We begin our examination of the types of objects that are counted by the Catalan numbers with paths, because the work has already been done. 1. 1 Cn counts the number of monotonic paths in an n  n grid that do not rise above the diagonal. □ # The Author 2015 S. 2. These are usually referred to simply as Dyck paths. 2 Cn counts the number of Dyck paths of length 2n that end on the horizontal axis. 3 Cn counts the number of 1) monotonic paths in an n  n grid that do not rise above the diagonal, 2) Dyck paths of length 2n that end on the horizontal axis.

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An Introduction to Catalan Numbers by Steven Roman


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