Update, and the combinatorics of quintic equations

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 [...]

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Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on [...]

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Extracting the diagonal

Needless to say, I have been very, very busy. But enough about me. Suppose you are given a bivariate generating function in “closed form,” where I’ll be vague about what that means. Such a generating function may arise, for example, from counting lattice paths in ; then might count the number of paths from to [...]

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The Catalan numbers, regular languages, and orthogonal polynomials

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 be the ordinary generating function for the ordered rooted trees on vertices (essentially we [...]

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