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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,

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.

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Running some experiments, it's clear that this sequence counts the number of ways to transpositionally invert a string,

*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.

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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.