SSL Minutes for Tuesday, Nov. 18, 2003

Notes: Sam Lachterman
Snacks: Martin Hock

Hal will take notes on Thursday.
Paul will bring snacks Thursday.

Jim: Have we found out about posting sequences to the Online Encyclopedia of 
Integer Sequences? 
Paul: Sam has done so--you fill out a form and then the moderator decides 
what gets posted.

Jim: A while back we found that for the sequence of matchings of the 
2-by-2-by-n (cube) snake, every other term is a perfect square (and the 
ones which are not are twice a perfect square).  Moreover, we showed that 
the weighted "version" is also a perfect square (or twice a square) for some
natural weighting scheme of the edges.  Has anyone looked into proving the
positivty of the coefficients?  That is, that they are counting something?

(No).

Jim: Well let's move on, if you all have moved on. Has anyone looked at the 
two articles which I sent in a prior e-mail about Markoff-like entities? 
That is, entities whose integer solutions can be found from a fundamental 
one?  Entities of the form: ax^2+by^2+cz^2=dxyz for constants a,b,c,d
Can we find a geometrical picture?

[Jim goes to board]
Given a triple (a,b,c) such that a<b<c and a^2+b^2+c^2=3abc
we can construct a new triple (a,b,c'), where c'=3ab-c=(a^2+b^2)/c
that has the same property.

Can we show that c' < c  ?
(No--showed c' < 2c).

Jim: Martin Isaacs has found a geometrical construction (which could possibly 
lead to exchange relations) which gives rational sides of the triangle (by 
using the circumcenter). He also has an idea which I have yet to look at.

I would also like everyone to start coming back to snakes.

[What have people been up to?]
Stephen: Working on my thesis.
Martin: Went to Papadimitrou's talk. Sam and I looked at papers by Reid Barton 
and David Speyer [which both dealt with combinatorial interpretations of 
certain bilinear recurrences.]
Hal: Didn't do much. Still interested in Soddy, cluster algebras, etc.
Paul: Why should we look at planar graphs of octagons and squares?
Jim: Good (that is, working) combinatorial model for certain interleaved 
(b,c)-sequences (Musiker and Propp) [pure algebraic proof given by Fomin 
and Zelevinsky].

[People begin talking about their confusions with Itsara paper.]
[Stephen will write to Andy Itsara about clarifications in paper.]

Carl: Looked at different kinds of snakes . . . Most reduce to square snakes.
Abby: Looked at matchings of triangles and square snakes. [Suggests she 
understands issue in Itsara paper--will e-mail explanation.]

Emilie: Looked at weightings and transfer matrices--nothing to report on this 
front. 

The (b,c)-sequence in the cluster algebra literature refers to

s_n = ((s_(n-1))^b + 1)/s_(n-2) for n odd
      ((s_(n-1))^c + 1)/s_(n-2) for n even

Results known for bc <= 4.

Stephen: Would like to work on combinatorial models for the (b,c)-sequences.
Martin: Interested in expander graphs viz a viz e-mail from Jim.
Jim: It might be fruitful to look at Markoff graphs modulo a prime.
Hal: Ok, *that* interests me. How far can we generalize the Markoff relation? 
(Soddy, etc.)

Hal would like to point out the "cube" shape of the Markoff tree modulo 3. 
(Anything further?)