March 2nd, 2007

beartato phd

(no subject)

I was doodling during a talk about a guy's research that I didn't really understand after the first few slides, and it occurred to me staring at my doodlings that for any n, the tilings of a regular 2n-gon (of side length 1) by rhombuses (of side length 1) with angles of 180k/n degrees for some integer k ought to be in a nice bijective correspondence with the number of different ways you can rewrite a string of length n to the reverse of that string by sequentially transposing adjacent characters, but modded out by a sort of concurrent equality on rewritings that seems very CLF-like. Here is a picture, showing one way to rewrite abcde (which occurs down the left side of the diagram) to edcba (which occurs down the right side):

Every edge labelled with a particular letter has the same orientation; each rhombus is kind of like a commutative-diagram 2-cell asserting the commutativity of two letters. The "concurrent equality" says that you should consider equal the two otherwise different ways of rewriting abcde to baced; the one that transposes ab first, and the other that transposes de first. Since they don't interfere, they're equal.

One cute thing in the case of n=5 is that the rhombuses involved are Penrose tiles. Dunno if this has anything to do with anything.


Running some experiments, it's clear that this sequence counts the number of ways to transpositionally invert a string, without the crazy concurrent equality — which I guess corresponds to some sort of ordered rhombus-tiling of a 2n-gon.