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Catalan Numbers: A Magical Sequence in Many Guises
The Catalan numbers are a sequence whose nth member is a particular sum of n products. They have proven to be ubiquitous with one scholar identifying some 190 situations in which they apply! The Catalans were discovered by Minggatu by 1730 and Euler in the 1750s long before Eugène Charles Catalan [masked]) did his work on them. It was a scholar around 1900 who introduced the misattribution, so we should probably call them Minggatu numbers. This event will feature an depth exploration of the Catalans which we will define 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) where C(0)=1 and n is a nonnegative integer. Here is a set of 13 exercises to guide your preparation and participation: . The topic is largely based on three primary video lectures from Ma Yuchun's free on-line TsinghuaX course "Combinatorial Mathematics" (see ). The three main lectures are from the Week 5 unit on "Magical Sequences" in the subunit "Catalan Numbers": 1. "Amazing World of Computers": My notes (includes links to the video on YouTube and subtitles): 2. "Catalan Numbers": My notes: 3. "Examples of Catalan Numbers": My notes: In addition to support our on-going series of topics with a generating functions theme (one can derive a formula for computing the Catalans using generating functions), the topic will also feature George Pólya's paper "On Picture-Writing" which provides an introductory treatment of the method of generating functions. You can obtain free access to Pólya's paper 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]. For those participants who are particularly interested in the generating functions aspect of this topic, I would like to highlight questions #10, #11, and #12 in the exercise set which feature generating functions: . We will spend a lot of time making sure participants understand those three questions during the event. For those who might benefit from background material on the basic sum and product rules for counting, the basic idea of combinations including the notation for combination numbers or binomial coefficients, and the lattice path model for combinations, these three 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 ● "Models of Combination and Combinatorial Identities" in the subunit "Permutation or Combination?": My notes . For those who might benefit from additional background material on generating functions here are references to a free on-line book and three Ma Yuchun lectures from her Week3 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 . ● "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 . I do not recommend consulting the following resources (they are unnecessary and will require too much time to review), here are links to the previous five events Math Counts has organized on Ma Yuchun's course: Sep 2015: Combinatorics: The Science of Counting and Arrangements (, May 2016: Combinations and Permutations (, Oct 2016: Counting with Generating Functions (Integer Partitions and more) (, Jan 2017: Solving Basic Recurrence Relations with Generating Functions (, and Feb 2017: Solving Linear Homogeneous Recurrence Relations (

The Corner Bakery Cafe

17th and JFK Blvd · Philadelphia, PA


What we're about

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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