The shuffle pasting, I

Sjoerd Crans, CTRC Seminar 11 April 2000

This is the first in a series of talks investigating the shuffle pasting. The goal is to show that shuffles form a well-formed loop-free pasting scheme, and to characterize well-formed subpasting schemes.

Motivation: for braided monoidal 2-categories a partial coherence theorem, for Zamolodchikov systems need the shuffle pasting as part of permutohedra, for parity structures on permutohedra and associahedra, for looping of pasting schemes which could formulate axioms for teisi, and for this is interesting mathematics.

I gave several ways to describe n-dimensional (p,q)-shuffles, the most important one being as a pair of functions f: p+q -> {0,1} and g: p+q -> p+q-n. The idea is that of a sequence of p 0's and q 1's with some (01)'s bracketed. Such a bracketed pair is called a swap.

The (n-1)-dimensional faces of such an n-dimensional (p,q)-shuffle are given by removing one pair of brackets and replacing it by 01 or 10 depending on whether there is an even or odd (for source, odd or even for target) number of swaps before the pair under consideration. As such, any n-dimensional (p,q)-shuffle has the shape of an n-cube.

See sections 1-3 of the paper ``The shuffle pasting'' (in preparation).

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