## November 30, 2019

### Random Permutations (Part 5)

#### Posted by John Baez

To go further in understanding the properties of random permutations, we need to use ‘bivariate’ generating functions — that is, generating functions involving 2 variables. Here’s a good introduction to these:

- Phillipe Flajolet and Robert Sedgewick,
*Analytic Combinatorics*, Part III: Combinatorial Parameters and Multivariate Generating Functions.

I will try to explain these in a way that takes advantage of Joyal’s work on species. Then I’ll use them to tackle this puzzle from Part 3.

**Puzzle 5.** What is the expected number of cycles in a random permutation of an $n$-element set?

## November 25, 2019

### Random Permutations (Part 4)

#### Posted by John Baez

Last time I listed a bunch of facts designed to improve our mental picture of a typical randomly chosen permutation. Many of these facts are fun to prove using generating functions. But one has a more elementary proof.

Namely: a random permutation of a large finite set has an almost 70% chance of having a single cycle that contains at least half the elements!

This really solidifies my image of a typical permutation of a large finite set. It consists of one big cycle involving most of the elements… and the rest, which is a typical permutation of the remaining smaller set.

## November 24, 2019

### Random Permutations (Part 3)

#### Posted by John Baez

I’m trying to understand what a random permutation of a large set is typically like. While I’ve been solving some puzzles that shed a bit of light on this question, I now realize there are some other puzzles that would help more!

### Topics in Category Theory: A Spring School

#### Posted by Tom Leinster

*guest post by Emily Roff*

**Topics in Category Theory: A Spring School** will be taking place on March 11th–13th 2020 at the International Centre for Mathematical Sciences (ICMS)
in Edinburgh, Scotland. The idea of the meeting is to gather together PhD
students and junior researchers who use category-theoretic ideas or
techniques in their work, and to provide a forum to learn about important
themes in contemporary category theory, from experts and from each other.

### Random Permutations (Part 2)

#### Posted by John Baez

Here’s a somewhat harder pair of problems on random permutations, with beautiful answers.

### Random Permutations (Part 1)

#### Posted by John Baez

I’m going to solve some problems on random permutations in my combinatorics class, to illustrate some of the ideas on species and generating functions, but also to bring in some ideas from complex analysis.

In all these problems we choose permutations randomly from $S_n$, with each permutation having probability $1/n!$ of being chosen.

I’ll start with one that’s easy, given the stuff I’ve already explained in class.

## November 23, 2019

### The Golomb-Dickman Constant

#### Posted by John Baez

I’m falling in love with random permutations. There’s something both simple and fairly deep about them. I like to visualize a random permutation as a “gas of cycles”, governed by the laws of statistical mechanics. I haven’t gotten very with this analogy yet. But I’m learning a lot of fun stuff.

## November 19, 2019

### Chair in Pure Mathematics at Sheffield

#### Posted by Simon Willerton

Here at the University of Sheffield we are advertising a Professorship in Pure Mathematics. The application deadline is 12 January 2020.

The research in pure maths is clustered around Topology, Number Theory, and Algebra and Geometry. People with interests allied to those of the Café are strongly encouraged to apply!

Please don’t hesitate to contact Sarah Whitehouse or me if you have any questions or would like to chat informally about this position.

## November 18, 2019

### Total Maps of Turing Categories

#### Posted by John Baez

*guest post by Adam Ó Conghaile and Diego Roque*

We continue the 2019 Applied Category Theory School with a discussion of the paper Total maps of Turing categories by Cockett, Hofstra and Hrubeš. Thank you to Jonathan Gallagher for all the great help in teaching our group, to Pieter Hofstra for suggesting and coordinating the project and to Daniel Cicala and Jules Hedges for running this seminar.

## November 16, 2019

### Geodesic Spheres and Gram Matrices

#### Posted by Tom Leinster

This is a short weekend diversion, containing nothing profound.

Any sphere in $\mathbb{R}^n$ can be given the geodesic metric, where the distance between two points is defined to be the length of the arc of a great circle between them. It’s not a big surprise that the same is true in any real inner product space $X$: for instance, the unit sphere $S(X)$ centred at the origin can be given a geodesic metric in a natural way. This defines a functor

$S: \mathbf{IPS} \to \mathbf{MS}$

from inner product spaces to metric spaces, where in both cases the maps are not-necessarily-surjective isometries.

What’s more interesting is that it’s not quite as straightforward to prove as you might imagine. One way to do it leads into the topic of Gram matrices, as I’ll explain.

## November 15, 2019

### Doubles for Monoidal Categories

#### Posted by John Baez

*guest post by Fosco Loregian and Bryce Clarke*

This post belongs to the series of the Applied Category Theory Adjoint School 2019 posts. It is the result of the work of the group “Profunctor optics” led by Bartosz Milewski, and constitutes a theoretical preliminary to the real meat, i.e. the discussion of Riley’s paper Categories of Optics by Mario Roman and Emily Pillmore (hi guys! We did our best to open you the way!)

In the following post, we first introduce you to the language of co/ends. Then we deconstruct this paper by Pastro and Street:

- Craig Pastro and Ross Street, Doubles for monoidal categories,
*Theory and Applications of Categories***21**4 (2008), 61–75.

Our goal is to make its main result (almost) straightforward: *Tambara modules* can be characterized as particular profunctors, precisely those that interact well with a monoidal action on their domain. For a fixed category $X$, these endoprofunctors form the Kleisli category of a monad on $Prof(X,X)$.