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Archive for the ‘combinatorics’ Category

Let C be a (locally small) category. Recall that any such category naturally admits a Yoneda embedding

\displaystyle Y : C \ni c \mapsto \text{Hom}(-, c) \in \widehat{C}

into its presheaf category \widehat{C} = [C^{op}, \text{Set}] (where we use [C, D] to denote the category of functors C \to D). The Yoneda lemma asserts in particular that Y is full and faithful, which justifies calling it an embedding.

When C is in addition assumed to be small, the Yoneda embedding has the following elegant universal property.

Theorem: The Yoneda embedding Y : C \to \widehat{C} exhibits \widehat{C} as the free cocompletion of C in the sense that for any cocomplete category D, the restriction functor

\displaystyle Y^{\ast} : [\widehat{C}, D]_{\text{cocont}} \to [C, D]

from the category of cocontinuous functors \widehat{C} \to D to the category of functors C \to D is an equivalence. In particular, any functor C \to D extends (uniquely, up to natural isomorphism) to a cocontinuous functor \widehat{C} \to D, and all cocontinuous functors \widehat{C} \to D arise this way (up to natural isomorphism).

Colimits should be thought of as a general notion of gluing, so the above should be understood as the claim that \widehat{C} is the category obtained by “freely gluing together” the objects of C in a way dictated by the morphisms. This intuition is important when trying to understand the definition of, among other things, a simplicial set. A simplicial set is by definition a presheaf on a certain category, the simplex category, and the universal property above says that this means simplicial sets are obtained by “freely gluing together” simplices.

In this post we’ll content ourselves with meandering towards a proof of the above result. In a subsequent post we’ll give a sampling of applications.

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The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let G be a finite p-group acting on a finite set X. Let X^G denote the subset of X consisting of those elements fixed by G. Then |X^G| \equiv |X| \bmod p; in particular, if p \nmid |X| then G has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

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Previously I mentioned very briefly Granville’s The Anatomy of Integers and Permutations, which explores an analogy between prime factorizations of integers and cycle decompositions of permutations. Today’s post is a record of the observation that this analogy factors through an analogy to prime factorizations of polynomials over finite fields in the following sense.

Theorem: Let q be a prime power, let n be a positive integer, and consider the distribution of irreducible factors of degree 1, 2, ... k in a random monic polynomial of degree n over \mathbb{F}_q. Then, as q \to \infty, this distribution is asymptotically the distribution of cycles of length 1, 2, ... k in a random permutation of n elements.

One can even name what this random permutation ought to be: namely, it is the Frobenius map x \mapsto x^q acting on the roots of a random polynomial f, whose cycles of length k are precisely the factors of degree k of f.

Combined with our previous result, we conclude that as q, n \to \infty (with q tending to infinity sufficiently quickly relative to n), the distribution of irreducible factors of degree 1, 2, ... k is asymptotically independent Poisson with parameters 1, \frac{1}{2}, ... \frac{1}{k}.

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Previously we showed that the distribution of fixed points of a random permutation of n elements behaves asymptotically (in the limit as n \to \infty) like a Poisson random variable with parameter \lambda = 1. As it turns out, this generalizes to the following.

Theorem: As n \to \infty, the number of cycles of length 1, 2, ... k of a random permutation of n elements are asymptotically independent Poisson with parameters 1, \frac{1}{2}, ... \frac{1}{k}.

This is a fairly strong statement which essentially settles the asymptotic description of short cycles in random permutations.

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The following two results are straightforward and reasonably well-known exercises in combinatorics:

  1. The number of permutations on n elements with no fixed points (derangements) is approximately \frac{n!}{e}.
  2. The expected number of fixed points of a random permutation on n elements is 1.

As it turns out, it is possible to say substantially more about the distribution of fixed points of a random permutation. In fact, the following is true.

Theorem: As n \to \infty, the distribution of the number of fixed points of a random permutation on n elements is asymptotically Poisson with rate \lambda = 1.

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Previously we described all finite-dimensional random algebras with faithful states. In this post we will describe states on the infinite-dimensional ^{\dagger}-algebra \mathbb{C}[x]. Along the way we will run into and connect some beautiful and classical mathematical objects.

A special case of part of the following discussion can be found in an old post on the Catalan numbers.

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A brief update. I’ve been at Cambridge for the last week or so now, and lectures have finally started. I am, tentatively, taking the following Part II classes:

  • Riemann Surfaces
  • Topics in Analysis Probability and Measure
  • Graph Theory
  • Linear Analysis (Functional Analysis)
  • Logic and Set Theory

I will also attempt to sit in on Part III Algebraic Number Theory, and I will also be self-studying Part II Number Theory and Galois Theory for the Tripos.

As far as this blog goes, my current plan is to blog about interesting topics which come up in my lectures and self-study, partly as a study tool and partly because there are a few big theorems I’d like to get around to understanding this year and some of the material in my lectures will be useful for those theorems.

Today I’d like to blog about something completely different. Here is a fun trick the first half of which I learned somewhere on MO. Recall that the Abel-Ruffini theorem states that the roots of a general quintic polynomial cannot in general be written down in terms of radicals. However, it is known that it is possible to solve general quintics if in addition to radicals one allows Bring radicals. To state this result in a form which will be particularly convenient for the following post, this is equivalent to being able to solve a quintic of the form

\displaystyle y = 1 + xy^5

for y in terms of x. It just so happens that a particular branch of the above function has a particularly nice Taylor series; in fact, the branch analytic in a neighborhood of the origin is given by

\displaystyle y = \sum_{n \ge 0} \frac{1}{4n+1} {5n \choose n} x^n.

This should remind you of the well-known fact that the generating function \displaystyle y = \sum_{n \ge 0} \frac{1}{n+1} {2n \choose n} x^n for the Catalan numbers satisfies y = 1 + xy^2. In fact, there is a really nice combinatorial proof of the following general fact: the generating function \displaystyle y = \sum_{n \ge 0} \frac{1}{(k-1)n+1} {kn \choose n} x^n satisfies

y = 1 + xy^k.

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