## SSL Minutes

1 May 2001

Kristin Jehring

Geir will bring beverages for Thursday, 05/03.

We got off track for about the first 15 minutes, then we went outside.

We briefly discussed the homework. Those who did it saw that Jim's conjecture holds for order 4. Jim hopes that the Razumov/Stroganov conjectures lead to a general formula which will give the total number of ASMs for a specific pairing.

We will have SSL meetings May 15 and 17. The final dinner on the 17th will close up this semester's SSL. The dinner is set to be at the Wilson Street Grill, unless someone objects. We agreed to go half with Jim on our dinner.

After the final meeting, you can still meet with Jim or email him. However, he is moving at the end of May.

Putting stuff on the SSL webpage: email your final, stable documents to Jim so it can be put permanently onto the webpage.

We need more open questions, so we brainstormed on this for awhile. For Nick to put on page:

- A(n-1) conjecture
- ASM shuffling conjecture
- How many different ASMs correspond to a given pairing of a DPFL?
Jim mentioned a bunch of questions he wants us to think about and try to answer in these last two weeks. Some of these were answered as they were brought up.

- How many different paths connect 1-->2 in a DPEL? Related question, can you get any length?
- What is the longest path that connects 1-->2?
- What is the average number of loops in a DPFL? What are the lengths and shapes of the loops?
- If n is odd, can have path that connects the midpoint of two opposite sides in a DPFL. How many ASMs correspond to this specific pairing?

(Jim had a thought about Catalan objects: Know the average number of components is 3 at the limit. Look for stats of Catalan objects which might lead to interesting characteristics of ASMs)- Can a permutation matrix be reconstructed from its pairing? ANS: NO!
- Given a permutation matrix of order n, how many ASMs of n+1 are compatible with it?
- In the ASM-compatibility tree, are there as many ways to go up&down as there are ways to go down&up?
- What is the 'refined' A(n-1) conjecture? [what happens to the 1]
- For lozenge tilings of hexagons, try looking at non-regular hexagons of the form (a,b,c). Try setting a,b constant.
- If you are counting things, look at the numbers modulo some smaller number. Try to find 2-adic properties.
- Decorating Dominoes: Connect the six vertices of a domino in some way and tile a region with these decorated dominoes, try to create paths across the dominoes.
- Do you see patterns in the patterns?
Jim urges us to expand our minds. Try and come up with new business methods. Create a new way of looking at a problem or come up with another problem from one you are working on. Can't solve a problem, try a simpler one.

Jim got a phone call.

## Progress Reports:

Halworked on writing up his proof. has sent his paper to the list. Would like to have feedback. Also got his haircut.

Nick & Abeworked on program that would create the ASM-compatability tree.

Racheldidn't do much over the weekend. Has T-shirt design made, just needs to compile. Also, need to send the design to the list to get feedback. Will call t-shirt places to find out costs and such things.

Dominiclooking at the 2n+1 ASM of TOAD. Interested in the reflections and rotations of the non-Baxter permutation matrices.

Boytchoworked on HW a little, need to digitize the T-shirt design.

Kristinworked on HW, emailed data, got more data, stuck on proving her conjecture.

Geirread Hal's paper and the Razumov and Stroganov paper.

Danworked on Hal's applet, added the features Jim asked for, program has a few bugs now though.

Jimplayed with nests in DPFLs