## SSL Minutes

6 March 2001

Abe and Joel

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.

## 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.