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Minggatu Numbers and Generating Functions; Catalan's Triangle
The sequence of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... is often called the Catalan or Segner numbers. Since Minggatu discovered them first (by 1730), we call them Minggatu numbers. They have proven to be ubiquitous with one scholar identifying some 190 situations in which they apply! We define them by Johann Andreas Segner's recurrence relation C(n)=C(0)C(n-1)+C(1)C(n-2)+···+C(n-2)C(1)+C(n-1)C(0)=Σ(from k=0 to n-1) C(k)*C(n-k-1) where C(0)=1 and n is a nonnegative integer. This event will use Catalan's Triangle to further introduce the Minggatu numbers. Then extensive application of the method of generating functions will be used to further explore the Minggatus. Two important applications will be considered: the number of distinct triangulations of an n-gon and the number of distinct trees with n+1 vertices. These problems will reveal more about the magical relationships implicit in Minggatu numbers. Here is a set of 10 exercises to guide your preparation and participation: . The topic is based on a paper by George Pólya and two video lectures with Ma Yuchun. The first highlighted resource is George Pólya's paper "On Picture-Writing" which provides an introductory treatment of the method of generating functions. You can get access to it with a free MyJSTOR account at As a PA resident, I was able to obtain a PDF copy of the paper from Temple University. Because the paper is still under copyright, I cannot post it on-line. If you want a copy for educational purposes, send me an e-mail at [masked]. The two featured video lectures are from the free on-line TsinghuaX course "Combinatorial Mathematics" (see ) in the unit on "Magical Sequences" in the subunit "Catalan Numbers": 1. "Amazing World of Computers" (introduction to the Catalan or Minggatu numbers): My notes (which include links to the video on YouTube and subtitles): 2. "Catalan Numbers": My notes: For those who might want more background material on the basic sum and product rules for counting and the basic idea of combinations including the notation for combination numbers or binomial coefficients, these two additional Ma Yuchun video lectures from the TsinghuaX course unit on "Combinatorial trip of a Pingpang ball" should be helpful: ● "Fundamental Counting Principles" in the subunit 'Counting with "+" "-" "*" "/"': My notes: ● "Definition of Permutation and Combination" in the subunit "Permutation or Combination?": My notes at For those who might want additional background material on generating functions here are references to a free on-line book and three Ma Yuchun lectures from her unit "Generating Function": ● See the first four sections of chapter 4 on "Generating Functions" of Kenneth Bogart's free on-line book "Combinatorics Through Guided Discovery" at . In addition §1.2 includes material on the basic counting rules and §1.3.1 introduces the Catalan (Minggatu) numbers. ● "Definition Of Generating Function (1)": My notes at . ● "Definition Of Generating Function (2)": My notes at . ● "What Can We Do With Generating Function": My notes at .

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Math Counts is a meetup to engage all things mathematical in a relaxed setting on the fourth Saturday of each month. We strive to make each event accessible to those with rudimentary math skills while also engaging those with more advanced knowledge, so join us no matter what your level of mathematical ability.

Math Counts brings together math aficionados, amateur and professional mathematicians and educators to engage all things mathematical. Our meeting topics range from the elementary to the profound, the practical to the philosophical, and the simple to the complex. Whether we are discussing books or on-line videos, hanging out to discuss recent mathematics news, enjoying mathematics activities, or otherwise imbibing the mathematical, we invite you to join us in a relaxing setting for stimulating polite conversations and activities to participate in the fabric of Philadelphia's vibrant Mathematics tapestry.

Mathematics is surprising, playful, stimulating and profoundly applicable to most aspects of life. Keith Devlin and others call it the science of patterns. Here are some quotes about the subject:

"If the modern world stands on a mathematical foundation, it behooves every thoughtful, educated person to attempt to gain some familiarity with the world of mathematics. Not only with some particular subject, but with the culture of mathematics, with the manner in which mathematicians think and the manner in which they see this world of their own creation."
--- William Byers

"A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." --- An Old French Mathematician quoted by David Hilbert

"But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident."
--- Felix Klein

"I assert only that in every particular Nature-study, only so much real science can be encountered as there is mathematics to be found in it"
--- Immanuel Kant

"The greatest challenge today, not just in cell biology and ecology but in all of science, is the accurate and complete description of complex systems. Scientists have broken down many kinds of systems. They think they know most of the elements and forces. The next task is to reassemble them, at least in mathematical models that capture the key properties of the entire ensembles."
--- E. O. Wilson, Consilience, p.85.

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