Graph Theory

Steven Roman's An Introduction to Catalan Numbers PDF

By Steven Roman

ISBN-10: 3319221434

ISBN-13: 9783319221434

This textbook offers an creation to the Catalan numbers and their striking homes, besides their a variety of functions in combinatorics. Intended to be available to scholars new to the topic, the ebook starts off with extra simple issues ahead of progressing to extra mathematically subtle topics. Each bankruptcy specializes in a particular combinatorial item counted by means of those numbers, together with paths, bushes, tilings of a staircase, null sums in Zn+1, period constructions, walls, diversifications, semiorders, and more. Exercises are incorporated on the finish of publication, in addition to tricks and recommendations, to assist scholars receive a greater clutch of the material. The textual content is perfect for undergraduate scholars learning combinatorics, yet also will entice a person with a mathematical history who has an curiosity in studying concerning the Catalan numbers.

“Roman does an admirable activity of delivering an creation to Catalan numbers of a distinct nature from the former ones. He has made an exceptional number of subject matters so that it will express the flavour of Catalan combinatorics. [Readers] will collect a superb feeling for why such a lot of mathematicians are enthralled by way of the awesome ubiquity and style of Catalan numbers.”

 - From the foreword through Richard Stanley

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In particular, w has the form w ¼ αðβÞγ ð7:1Þ where α, β, and γ are fully parenthesized words, and lenðβÞ > 1 and the parentheses shown are the first matching pair, that is, the open parenthesis is the first open parenthesis in w and the closing parenthesis matches the open parenthesis. Note first that one of α or γ must be the empty word, since otherwise w is not fully parenthesized, for it is not possible to further parenthesize this expression in # The Author 2015 S. 1007/978-3-319-22144-1_7 39 40 7 Catalan Numbers and Algebraic Widgits order to group α and β together or β and γ together without introducing an open parenthesis somewhere to the left of the first open parenthesis.

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.

K. In other words, w is completely determined by these two sequences. For example, let n ¼ 6 and consider the count sequences ðα1 ; α2 ; α3 Þ ¼ ð3; 5; 6Þ and ðβ1 ; β2 ; β3 Þ ¼ ð1; 3; 4Þ Then the new-A positions are ðp1 ; p2 ; p3 Þ ¼ ð4; 8; 10Þ and a fill-in-the-gaps construction will construct the unique word w with these count sequences. First, we insert the new As where they belong: w¼ A A A 1 2 3 4 5 6 7 8 9 10 11 12 Then we compute the number of As between consecutive new As (and before the first new A and after the last new A): α1 À 1 ¼ 2 α2 À α1 À 1 ¼ 1 α3 À α2 À 1 ¼ 0 n À α3 ¼ 0 Now we can fill in the gaps to get the Catalan word: w¼AABAABBABA B B 1 2 3 4 5 6 7 8 9 10 11 12 It is clear that both count sequences α1 Á Á Áαk and β1 Á Á Áβk are strictly increasing because each gap between new As must include at least one B.

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An Introduction to Catalan Numbers by Steven Roman

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