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Posts Tagged ‘generating functions’

Kraft’s inequality: Let L be a prefix code on an alphabet of size k, and let l_n denote the number of codewords of length n. Then \displaystyle \sum_{n \ge 1} \frac{l_n}{k^n} \le 1.

In lieu of reposting my old blog posts I thought I’d revisit some of their topics instead. In an old post of mine I set this inequality as a practice problem. I had an easy but uninspired proof in mind, but recently I learned that this problem is an exercise in the first chapter of The probabilistic method by Alon and Spencer. The probabilistic proof is quite elegant, so I thought it was worth presenting.

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In the previous post we used the Polya enumeration theorem to give a sneaky, underhanded proof that

\displaystyle \sum_{m \ge 0} Z(S_m) t^m = \exp \left( z_1 t + \frac{z_2 t^2}{2} + \frac{z_3 t^3}{3} + ... \right).

If you’ve never seen the exponential function used like this, you might be wondering how it can be “explained.”

To explore this question, I’d like to give three other proofs of this result, the last of which will be “direct.” Along the way I’ll be attempting to describe some basic principles of species theory in an informal way. I’ll also give some applications, including to a Putnam problem.

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I ended the last post by asking whether the proof of baby Polya extends to the multi-parameter setting where we want to keep track of how many of each color we use. In fact, it does. First, we should specify what exactly we’re trying to compute. Recall the setup: we have colors 1, 2, ... n (represented by variables r_1, r_2, ... r_n), and we have a set of slots S with |S| = m acted on by a group G where each slot will be assigned a color. Define f_G(t_1, ... t_n) to be the number of orbits of functions S \to \{ 1, 2, ... n \} under the action of G where color i is used t_i times. (Since the action of G only permutes slots, it doesn’t change the multiset of colors used.) What we want to compute is the generating function

\displaystyle F_G(r_1, r_2, ... r_n) = \sum_{t_1 + ... + t_n = m} f_G(t_1, t_2, ... t_n) r_1^{t_1} r_2^{t_2} ... r_n^{t_n}.

Note that setting r_1 = r_2 = .... = r_n = 1 we recover F_G(n), which doesn’t contain information about particular color combinations. By the orbit-counting lemma, this is equivalent to computing

\displaystyle \frac{1}{|G|} \sum_{g \in G} | \text{Fix}(g) |(r_1, r_2, ... r_n)

where we must now count fixed points in a weighted manner, recording the multiset of colors in each fixed point. How do we go about doing this?

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We’ve got everything we need to prove the Polya enumeration theorem. To state the theorem, however, requires the language of generating functions, so I thought I’d take the time to establish some of the important ideas. It isn’t possible to do justice to the subject in one post, so I’ll start with some references.

Many people recommend Wilf’s generatingfunctionology, but the terminology is non-standard and somewhat problematic. Nevertheless, it has valuable insight and examples.

I cannot recommend Flajolet and Sedgewick’s Analytic Combinatorics highly enough. It is readable, includes a wide variety of examples as well as very general techniques, and places a great deal of emphasis on asymptotics, computation, and practical applications.

If you can do the usual computations but want to learn some theory, Bergeron, Labelle, and Laroux’s Combinatorial Species and Tree-like Structures is a fascinating introduction to the theory of species that requires fairly little background, although a fair amount of patience. It also contains my favorite proof of Cayley’s formula.

Doubilet, Rota, and Stanley’s On the idea of generating function is part of a fascinating program for understanding generating functions with posets as the fundamental concept. I may have more to say about this perspective once I learn more about it.

While it is by no means comprehensive, this post over at Topological Musings is a good introduction to the basic ideas of species theory.

And a shameless plug: the article I wrote for the Worldwide Online Olympiad Training program about generating functions is available here. I tried to include a wide variety of examples and exercises taken from the AMC exams while focusing on techniques appropriate for high-school problem solvers. There are at least a few minor errors, for which I apologize. You might also be interested in this previous post of mine.

In any case, this post will attempt to be a relatively self-contained discussion of the concepts necessary for understanding the PET.

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I’ve been inspired by The Unapologetic Mathematician (and his pages and pages of archives!) to post more often, at least for the remainder of the summer. So here is a circle of ideas I’ve been playing with for some time.

Let C(x) = \sum_{n \ge 0} C_n x^n be the ordinary generating function for the ordered rooted trees on n+1 vertices (essentially we ignore the root as a vertex). This is one of the familiar definitions of the Catalan numbers. From a species perspective, ordered rooted trees are defined by the functional equation

\displaystyle C(x) = \frac{1}{1 - xC(x)}.

The generating function \frac{1}{1 - x} = 1 + x^2 + ... describes the species \textsc{Seq} of sequences. So what this definition means is that, after tossing out the root, an ordered rooted tree is equivalent to a sequence of ordered rooted trees (counting their roots) in the obvious way; the roots of these trees are precisely the neighbors of the original root. Multiplying out gets us a quadratic equation we can use to find the usual closed form of C(x), but we can instead recursively apply the above to obtain the beautiful continued fraction

\displaystyle C(x) = \frac{1}{1 - \frac{x}{1 - \frac{x}{1 - ...}}}.

Today’s discussion will center around this identity and some of its consequences.

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