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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Newton’s sums, necklace congruences, and zeta functions

The goal of this post is to give a purely combinatorial proof of Newton’s sums which would have interrupted the flow of the previous post. Recall that, in the notation of the previous post, Newton’s sums (also known as the first Newton-Girard identity) state that . One way to motivate a combinatorial proof is to [...]

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GILA VI: The cycle index polynomials of the symmetric groups

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

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