From propp@math.wisc.edu Sat Feb  9 10:36:48 2002
Date: Mon, 10 Dec 2001 20:03:54 -0600 (CST)
From: propp@math.wisc.edu
To: reach@math.harvard.edu
Subject: [reach] minutes

Reach Minutes - 11/27/01 (prepared by Billy Hillegass and lightly edited
by Jim Propp)

Visit From NSF

The NSF seemed more interested in the math department in general than
the REACH program.

Logistics

+ Prof. Propp wants to take Reach participants out to dinner.

+ The dinner is scheduled for Monday 10th at 6:30 at Bertucci's.

+ Prof. Propp suggest abandoning the 5 minute format on Tuesdays, in
  favor of a variable length discussion of different problems.

+ Prof. Propp asks who is interested in participating next year.

+ The meetings for next semester might start 30 minutes earlier.

Infrastructure

+ Efstathios is going to set up a way for people to know what others
  are working on.

+ Federico wants a program to simplify maple output. e.g.

  x  y  z  + z  z  z  +  z  x  y  becomes
   0  0  2    0  1  2     0  1  1

  0   0           1   0           0   1
  1   1   1   ,   0   0   1   ,   1   0   0
  0   0           1   0           0   1

Agenda

+ Routings: Bijection or Lattice paths of (+1,0), (+1,+1),(-1,-1) type

+ Lionel: Stirling numbers

+ Bridget: Grids of rational functions

+ David: Reciprocity

Routing Bijections

Prof. Propp showed a bijection between domino tilings of a nxm 
rectangle, triangle lattice paths, and perfect matchings.

Grids of Rational functions

        1
-----------------    generates the grid
             2  2
1 - x - y - x  y

1   5  18  47 101
1   4  12  26  47
1   3   7  12  18
1   2   3   4   5
1   1   1   1   1

The coefficients are almost polynomials.  There are errors introduced by
the boundary.

The errors propagate at fastest linearly.

Generalized Stirling numbers

Lionel and David proved

sum  A(n,k) x^n y^k / n!  =  exp (y sum |G_n| x^n / n!).

where A(n,k) is the Stirling number for the Group G_n.

The Stirling number depends only on the order of the group.

Reciprocity

A sequence produced by concatenating a graph with n vertices on the edge
produces a sequence of 2^n by 2^n matrices 
        A^1 A^2 A^3 A^4 A^5 A^6...
Reciprocity corresponds to inverting these matrices.

This result proves the bowtie recurrence.

New Problem in Reciprocity

Prof. Propp suggest investigating perfect matchings of sequences of 2
squares tilted at 45 degrees, and generalizations.