SSL Minutes

6 March 2001
Abe and Joel

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Sorry, Guys, I forgot about this... It was the week before spring break, so I must've gone brain-dead immediately afterward.

For the record, Joel had his version of the notes to me promptly... I'm the slow one.

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I.

Jim gave a brief discussion on cyclic permutations and groups:

Transformations in the group MUST be rotations in the plane, not reflections or anything else.

There was a discussion of this applied to rhombus tilings, which I won't even TRY to draw in ASCII. Rotations in this case can only be rotated 120 degrees, giving an orbit of order 3.

Say there are m tilings that are invariant under cyclic permutations and 3n tilings that aren't. The total number of permutations would be m+3n. Removing the redundant ones would give m+n, which is unlikely to have a nice answer.

A usual question: "What is the number of ... up to symmetry?"

Changing symmetry can often simplify equations. Additionally, transformations can be paired, etc.

II.

Another topic: Dense packing models of hexagon grids. Someone might want to look into this, as no one has investigated it yet.

III.

Notes on programming: IT may be more efficient to make straightforward programs without worrying about nice ways of packing bits and so forth. This will make modification and extension easier.

IV.

Coupling from the Past: It is difficult to generate a random ASM from scratch, but by repeatedly permuting in a random fashion, it will eventually reach something close to a uniform distribution. Coupling from the past is a method which is proven to accelerate this convergence.

V.

Question: What about convex hulls around a set of points? How close is it to a circle?

VI.

ASMs/TOADs: How does one create a dense packing from a TOAD? 1. superimpose two ASMs for TOAD 2. Turn these into Blue-green maps

We need some representation for TOADs and ASMs that will allow easy cycling through all variations. Semi-strict Gelfand Patterns are good for this application as they are easy to generate.

Date: Thu, 29 Mar 2001 22:59:54 -0600 (CST)
From: propp@math.wisc.edu
To: ssl@math.wisc.edu
Subject: correction to Notes for March 6

The notes for March 6 say:
>We need some representation for TOADs and ASMs that will allow easy
>cycling through all variations.  Semi-strict Gelfand Patterns are good
>for this application as they are easy to generate.

"Semi-strict Gelfand Patterns" should be replaced by "Monotone Triangles". Remember, Semi-strict Gelfand Patterns correspond to plane partitions; TOADs and ASMs correspond to monotone triangles. (Leastaways, ASMs correspond to monotone triangles; TOADS correspond to something a bit more complicated.)

Jim

technique: generation => convert to matching => examine

N is a function

\sum_A (1^{N(A)}) = number of ASMs
\sum_A (2^{N(A)}) = number of TOADs
\sum_A (3^{N(A)}) = number of nice function for some product formula?
\sum_A (4^{N(A)}) = UGLY (until nontrivial formula is proven)

Nice weighted product formulas are also important at times.

Next Time:

Q analogs and P-adic (P is prime)

that's all.

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