REACH is a combinatorics research group I'm in run by Jim Propp. I'll use this space to post things that I want to show oether memebers of the group, as a result, they will be even less polished than the rest of this site.

A summary of some Laurentness results and beginnings of a combinatorial interpretation for lattices of numbers every 3x3 submatrix of which has determinant 1. TeX DVI PS PDF

A Mathematica Notebook for playing with the Cube Recurence under cluster Algbebras. The first part is a verification of the cube recurrence: nice to see if you enjoy large polynomials cancelling. The second half is my atttempts at a weighted version. Mathematica Notebook

The first file is an ASCII drawing of a graph whose perfect matchings are counted by f(4,0,4) in Propp's 3-Dimensional Somos-3 recurrence. Expand your margins until it looks reasonable. The second file is the first several such graphes, including how to number the weights for any choices of n, i and j as long as 6*n+j isn't too big. The third file draws the first three relations of the recurrence on the same lattice so that you can see how they fit together. ASCII Graphic ASCII Graphic ASCII Graphic

Here are some notes that (I hope) explain the complete picture of the Somos-3 graphs of the proceeding paragraph. Plain Text

In order to investigate the Somos-3 problem in the previous note, I wrote two Mathematica notebooks that might be useful to others. The first contains the helpful function Degrees[poly,vars] which, given a polynomial and a list of variables returns all different combined degrees to which those variables occur. For example, Degrees[1+x^2+xy,{x,y}]={0,2}. It also implements the Somos recurrence. The second contains the function f[n,i,j], which returns all variable appearing in the corresponding Somos polynomial. It occasionally lists terms that don't actually appear, but it makes up for it by being able to handle 6*n+j as large as 100. It also contains X[N], which returns the terms appearing when 6*n+j=N under my normalizations and Y[N] which contains them organized by type of edge and sorted on the second coordinate. Mathematica Notebook Mathematica Notebook

This notebook computes the octahedron recurrence for any initial conidtion specified by a boolean function Initial[n,i,j]. f computes the weighted version and g the unweighted version. A[n] displays a sampling of values of g[n,i,j] in what is usually a good range of (i,j) to look at. Remember that f and g cache values, so you will need to rerun their definitions everytime you chanige Initial. The included values of inital have comments that indicate what family of graphs or "clasical" example this corresponds to. Mathematica Notebook

This is my continuing attempts to write up the method of cross-and-wrenches, which allows one to assosciate a family of graphs to any starting configuration of the octahedron recurrence. I will try to copy new drafts her as I write them. I welcome comments. There is one that is not worth sending however; I am using the symbols +, / and \ for certain purposes in my paper. I am aware they don't look very good or convey what they ought. I will eventually be defining new symbols for these pruposes using the picture environment, but i haven't gotten to that yet. These are place holders. LaTeX DVI PS PDF Just a note - candw.tex has many assosciated eps files that it needs to compile correctly. Unfortunately, the only copies I have are on my computer at home and I'm not sure I can get them to you might want to put a flag on the file indicating this until I get this dealt with.