Abstract
This chapter contains an introduction to generating functions. We present generating functions as a way to deal with recursive equations that appear in combinatorial problems. First, we define these functions and their basic properties, and give examples of many basic generating functions. Then, we show how they can be used to obtain a closed formula to the Fibonacci numbers, and how this technique extends to similar sequences. As a more complicated example, we show how to obtain a closed formula for the Catalan numbers. This is done both via generating functions and with a purely combinatorial approach. Then, we introduce the concept of the derivative, and how it can make manipulations of generating functions much easier. The last section explains why generating functions are called “functions”. Namely, we explain when it is possible to evaluate them at a point and when an actual function can be represented as a generating function. At the end of the chapter, we present 18 problems for the reader.
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Notes
- 1.
Although J.P.M. Binet is given the credit for this formula, it has been traced back to Euler in 1765 and de Moivre in 1730. See (for example) The Art of Computer Programming by D.E. Knuth for references.
- 2.
One can always do it by using complex numbers, but this is rarely needed in olympiad problems.
- 3.
By |y| we mean the absolute value of y. That is, |y|=y if y is non-negative, and |y|=−y if y is negative.
- 4.
We need to show that these Taylor series converge and behave well. However, the formal arguments needed for this are much simpler than what needs to be done otherwise. The interested reader may want to consider g(x) as this Taylor series and see that it satisfies \(\frac{g'(x)}{g(x)}=\frac {r}{1+x}\). Integrating on both sides and using the value of g(0) should give g(x)=(1+x)r (as a real function). Why are all these steps valid?
- 5.
Note that exp(−1) is the same number that seemed to appear in Example 1.4.6.
- 6.
The probability of an event is the ratio of the number of favorable cases to the number of total cases.
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Soberón, P. (2013). Generating Functions. In: Problem-Solving Methods in Combinatorics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0597-1_6
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DOI: https://doi.org/10.1007/978-3-0348-0597-1_6
Publisher Name: Birkhäuser, Basel
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