From propp@math.wisc.edu Sat Feb  9 10:36:36 2002
Date: Fri, 9 Nov 2001 14:50:40 -0600 (CST)
From: propp@math.wisc.edu
To: reach@math.harvard.edu
Subject: [reach] minutes

Minutes for 11/6, written by David Speyer

Bureaucracy
  Prof. Propp would like feedback on how REACH is running
    Drop it off anonymously (or not) in his mailbox
    Questions of especial interest are how to encourage collaboration, how
      coordination with 192 is working, anything else

  Efstathios will compile a list of what people are interested in working
    on, to aid in collaboration.

  He also has sent out a note on how to run Mathematica in idle time.

  Presentation for the NSF
    presentations will be at 2 on Friday (11/16), probably in the Math
      Library
    should last roughly 5 miunutes
    afterwards, informal discussion until 3, show up if at all possible
    Greg, Trevor and David seem to be the people available/interested to
      do presentations
   There will be practice presentations on Tues for REACH and Wed (11/17)
      for 192
   Prof Gross would like to talk to the presenters beforehand. It is not
      clear if this is possible as he will be away and hard to reach until
      Wednesday.
   Prof Propp will also be away this weekend, but available by email.

A problem that Prof. Propp would liek someone to work on: consider the
number of paths from (0,0) to (i,j) by steps of (0,2), (1,1) or (2,0). For
i or j fixed, this is a polynomial in the other variable. What happens for
i or j negative?  Specifically, can we get a good extension to both
negative?

Prof Propp points out that there is an explicit correspondence between
lattice paths from (0,0) to (i,j) and subsets of size i of an i+j element
set: just take the indices of the horizontal edges. We have reciprocity
for binomial coefficents both via negative sets and by running lattice
paths backwards. Do we get the same bijection?

Similarly, for domino tilings, there are three equivalent things:
matchings, routings, and spanning trees! It turns out that spanning trees
of a 2xn rectangle are in bijection with matchings of a 3x(2n-1) rectangle
wih a corner removed.  Once again, does the bijecion reverse?  There was 
some confusion about the proper indexing, which was resolved.

Trevor asked whether one could/should approach domino reciprocity via the
product of trig function formula. David pointed out that this would give
fractions. Professor Propp answered that in fact, for 2-adic reasons, one
should expect fractions. If A(n) is the number of tilings of a 2n x 2n
rectangle, we should apparently have A(-n)=2^(-2n+1) A(n-1). The first few
values should be A(-1)=1/2, A(-2)=1/4, A(-3)=9/8.

Trevor wants a fast algorithm to compute the determinant of a matrix with
nonzero diagonal terms.

Greg has been studying the recurrence g(n)=(g(n-1)^2+k)/g(n-2). Greg
conjectures g(n)=sum k^m (n-1+m choose n-1-m); g(n)=(2+k)g(n-1)-g(n-2).
g(n)=sqrt(k)h(n) where h are the polynomials on the first 192 problem set
with x=1/sqrt(K).

Stav has been computing paths through graphs of the form
  * - * - *
 / \ / \ / \
* - * - * - *
When he runs them backwards, he gets pictures that aren't paths!