Combinatorics

Date: Oct 25, 2010

How big are families with the same number of girls and boys?

Let's assume you wish to have children—moreover, you want just as many girls as boys. In fact, you are so very much determined that you set out to not stop getting kids until the number of girls matches precisely the number of boys. How big a family will that make you, on average? read more »

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Date: Mar 6, 2010

Computer proved monotonicity of some coefficients

Ming-Hua Lin from the University of Regina sent me the following problem:

Problem: Let p ≥ 2 be an integer, and define

$$ f (t) = \left( \frac{1}{1 - t \left( \frac{1}{p} + \frac{p - 1}{2 p^2} t<br /> \right)} \right)^p = \sum_{n \geqslant 0} c_n t^n . $$

Show that $ c_2 > c_3 > c_4 > \cdots $. read more »

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Date: Jun 10, 2009

Fast q-binomials in Mathematica

Recently, I have been doing experiments involving q-binomial coefficients in Mathematica. Starting with version 7, Mathematica is prepared for some q-business; in particular, there exists a function named QBinomial giving the q-analog of Binomial. However, this implementation turned out to not be fast enough for my needs. Here is an alternative approach which is not only way faster but provides a full factorization. read more »

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Date: Apr 23, 2007

Eulerian numbers

This short introduction to Eulerian numbers accompanied my talk for the WZ Theory class held by Tewodros Amdeberhan at Tulane University in Spring 2007. read more »

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euleriannumbers.pdf
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euleriannumbers-presentation.pdf
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