# The shuffle pasting, III

## Sjoerd Crans, CTRC Seminar 9 May 2000

This is the third 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.

As a refresher, I gave the four different ways to represent a (p,q)-shuffle and the relations between those, and the four different ways to represent the swaps for an n-dimensional (p,q)-shuffle and the relations between those. Then I gave an improved formula for the rank involving the n-kind of an n-dimensional (p,q)-shuffle, which tells between which 0's and 1's the swaps occur. Next, I introduced two partial orders: one on n-cells having a bounding (n-1)-kind in common, very much like the open triangle order for pasting schemes, and one on (n-1)-cells of the same (n-1)-kind, basically saying that 1's have moved forward and 0's backward, with the usual twist taking into account the parity of the swaps.

In classifying well-formed subpasting schemes, I will use two conditions on collections of n-cells relative to two collections of (n-1)-cells: filling and fitness. The strategy will be that well-formedness is implied by filling is implied by fitness is implied by well-formedness. In this talk I gave the definition of filling, which more or less says that the first partial order must be a total one, and I showed that filling implies well-formedness.

See sections 6-7 of the paper ``The shuffle pasting'' (in preparation).

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