From the-concourse-on-high Sat Feb  9 10:37:23 2002
To: reach@math.harvard.edu
Subject: Minutes for 9-25 (from Gregg)

Business:

-we should email Jim Propp with weekly progress reports on Fridays

-It is important to keep track of hours spent on Reach

-There is an email list reach@math.harvard.edu
-don't send big files on it, give people a webpage URL instead

-There will be an archive of files, email, and minutes on Jim's webpage

-An end of the term report will be important

-Let Jim know if you're not coming to a meeting

-Might want to check out www.math.wisc/~propp/tiling/, the domino forum or
the bilinear forum

-Study of bilinear recurence relations is a growing field, has
applications in physics and elliptical curves

-The reseach grant can help buy books for a library

-Jim will talk to Ruby and look into getting forms to avoid sales tax

-the note taker records promises like this, could = will

-Where is the money coming from:
	-NSF personal grant, through VIGRE, an REU supplement (NSF
encouraging vertical integration), and the NSA.  Non-US citizens might be
able to be funded via Jim's start-up funds at the University of Wisconsin

-people are looking into setting up X-windows so the TCL program will be
able to work.

-If you copy the graph TCL program into one's fas account, than you should
be able to use it on the computers of the Science Center Basement.  Also
the source code is on the web.

-Stirling reciprocity will be discussed next time





1st Problem from Monthly:
	David got a cubic/quintic.  Ethan got the same. Gregg got a
sequence of recurence relations.

-Ethan got a pair of third-order recurence relations, and we discussed how
this could be the same as 5 first-order by defining new functions h(n) =
f(n+1).  Generally a pair of third-order can be represented as 6
first-order.

-A computer can be used to encode possiblities instead of a brute force
search.





2nd Problem from Monthly: (A polyonimo spans a square if it can embedded
in the square touching all four sides.  There are six pentonimoes that
span the 3x3 square.  How many polyonimoes of (2n-1) cells span an nxn
square?)  The answer is known.  Deadline is Oct 31 for submission.

-Trevor: we can look at pentominoes as 3x3 matrices of 0's and 1's

-Jim: Can also look at how one can deform pentonimoes into each other via
simple shifts.

-David will email a coding method.

-Matt: half of the pentonimoes span the 3x3 square, and 1/2 of the
trionimoes span 2x2 square.  Maybe there's a pattern

-Trevor: thinks conjecture is false.






More about matching:

-Matt, David found recurence relations.

-Matt: let H(n) = # of matchings with 2n vertices in the middle. H(1) = 2,
H(2) = 14, H(n+1) = 6 H(n) - H(n-1) + 2 .

-Can do some algebra to get it in a homogeneous form:

	H(n+1) = 7 H(n) - 7 H(n-1) + H(n-2).

-Bridget: there is a nice bijection between matchings of G graph and H
graph when neither edge have parallel horizontal lines as matching.
However other case is more difficult to show.

-David, Alex suggest induction for 2nd case

-David's bijections: matching specified by where diagonals of matchings
are.  Horizontals will fall into place as needed.  Imagine sliding
matchings up and down diagonals.

-Alex thinks of a similar bijection where the matchings left and right
instead of sliding.

-There are other families of shapes to consider too.