*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.