15 May 2001
No one volunteered to bring drinks on Thursday.
Somebody asked about gyration on a Hex FPL.
Somebody asked Dominic what progress he has made on the Doubled-ASM FPL (The ``naturally interleaved'' kind). No numbers as yet.
Definition: A pairing is rigid if it has only one corresponding hex-tiling or ASM.
Definition: a pairing is locally rigid if there are no local moves that can be made which preserve the paring but change the tiling/ASM.
If there was a complete set of local moves for a given system, then the two definitions for rigidity would be equivalent.
That reminds me: I was going to look into the question of local moves.
Fun Summer Projects
Let me refer you to Chris Moore's email to the domino group.Date: Wed, 9 May 2001 14:09:40 -0600 (MDT)
From: Cris Moore <email@example.com>
To: domino nos@pam math.wisc.edu
Subject: ribbon tiles
Fellow domino'ers, you may enjoy finding bijective proofs of the following two things, which I'm thinking of writing a little note about when I get the time. Recall that a ribbon n-omino (or n-ribbon for short) is a path of n squares connected by stepping North or East (note rotations are not allowed). There are 2^(n-1) distinct n-ribbons.
Let N(x,y,n) be the number of tilings of x by y rectangles with n-ribbons. Then
1) for all n > x, N(x,kn,n) is a function only of x and k. For instance, there are exactly as many ways to tile a 3x8 rectangle with 4-ribbons as there are to tile a 3x10 rectangle with 5-ribbons.
2) for all n, N(n,n,n) = n!
Both of these have easy proofs which I'm sure you'll have no trouble finding... just remove diagonals of squares running NW-SE that each ribbon crosses exactly once, converting n-ribbons to (n-1)-ribbons.
- Cris Moore
Jim showed us the easy proof of item (2), then asked us to come up with some other questions. Here they are:
- How many ways to tile a 2n by 2n rectangle with n-ribbons? I came up with a lower bound of (2n)! and thought for a while that that might be an exact answer. But then Jim came up with (n!)^4 as a new lower bound.
- How many ways to tile a 2n by n rectangle with n-ribbons?
- How many ways to tile a l by m rectangle with n-ribbons?
- How many ways to tile a n+1 by n rectangle with n-ribbons?
We then talked about local moves on a ribbon tiling. I forget who invented these. Take two ribbons that lie next to each other for a while and swap their tails. They'll still be the same length, and take up the same space, but be in a different configuration.
What about square FPLs on a n-by-m grid? You would have to play with the boundary conditions for an odd-by-even grid. Jim said he had dealt with this in part 5 of "The many faces of ASMs". I think that he called them partial ASMs.
I then had an epiphany about hex FPLs. See my write-up.
Nick worked on a project for one of his classes.
Boytcho is writing up some stuff.
Abe studdied up for his exams. This week he'll do the glossary, and over the summer he'll learn about gyration.
Dominic has been obsessing about the sequence 16, 17, 20, 22, 24, 31, 40, 121. He's also been looking at the ``naturally interleaved'' ASMs, as mentioned before. He wants to modify TOAD Shuffler to show the funny lines mentioned in the notes from two weeks ago.
I have been busy with my paper.
Mike glanced at the eigenvalues of the R-S matrix again.
Jim went to a conference. He talked about Penrose tilings and how to produce one without mistakes by moving a pair of pentagons around the plane. Without benefit of a chalkboard, I didn't follow that very well.
Then we called it quits.