An Introduction to Catalan Numbers by Steven Roman

By Steven Roman

This textbook offers an advent to the Catalan numbers and their notable homes, in addition to their quite a few purposes in combinatorics. Intended to be available to scholars new to the topic, the publication starts with extra trouble-free subject matters ahead of progressing to extra mathematically refined topics. Each bankruptcy makes a speciality of a particular combinatorial item counted by means of 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 strategies, to assist scholars receive a greater snatch of the material. The textual content is perfect for undergraduate scholars learning combinatorics, yet also will entice a person with a mathematical heritage who has an curiosity in studying in regards to the Catalan numbers.

“Roman does an admirable task of offering an creation to Catalan numbers of a distinct nature from the former ones. He has made a great collection of issues with a purpose to express the flavour of Catalan combinatorics. [Readers] will collect a great feeling for why such a lot of mathematicians are enthralled through the extraordinary ubiquity and magnificence of Catalan numbers.”

 - From the foreword through Richard Stanley

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The nonprincipal blocks of P are the sets ! # J J&K If I and K are disjoint, then clearly so are BI and BK. If I and K are not disjoint, then we can assume that I & K, in which case I is one of the intervals that is removed in defining BK and so I \ BK ¼ ∅, which implies a fortiori that BI \ BK ¼ ∅. Thus, the family È  É P ¼ B I  I 2 I [ fR g is a partition of [n]. To see that P is noncrossing, suppose that 1 i

3. 4 with a ¼ 2) and so Dnþ2 ¼ Cn . 4 Cn counts the number of triangulations of a convex polygon with n þ 2 sides. □ Disk Stacking Sometimes it is easier to find a characterization of one type of object in terms of another type of object whose count we already know than to directly count the original objects. Here is an example. 9 shows one way to stack equal-sized disks in the plane, a task that we often find ourselves wishing to do. Let Dn be the number of possible disk stackings, where the bottom row has n disks.

Let P n be the family of all rooted chorded 2n-gons and let P n, k be the members of P n whose root vertex is adjacent to the vertex v2k, for 1 k n. 1. Note that if the root vertex is adjacent to either v2 or v2n, then either P‘ or Pr will be empty. In any case, if P‘ or Pr is nonempty, then it is properly chorded and we declare the root of P‘ to be the vertex v2 and the root of Pr to be the vertex v2n. Thus, for each 1 k n, we have an injective map θn, k : P n, k ! P kÀ1 Â P nÀk defined by θn, k ðPÞ ¼ ðP‘ ; Pr Þ The map θn,k is also surjective, since any two chorded rooted polygons can be recombined by the addition of a nexus chord (and concomitant vertices) to produce a new larger rooted chorded polygon.

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