The 2002 REACH tee-shirt

designed by Trevor Bass, with input from Gabriel Carroll
(for information on how to order, go to the bottom of this page)

The front of the tee-shirt shows the first few terms of the Somos-4 sequence: 1,1,1,1,2,3,7,23,59,314,...

If we write the nth term as a(n), the the formula giving each successive term of the sequence in terms of the preceding ones is

                a(n-1) a(n-3) + a(n-2)^2
        a(n) = --------------------------  .
So, for instance, to compute the next term after 314, we would compute (23*314+59*59)/7, which is 1529. It isn't obvious that this rule gives an infinite sequence of whole numbers, but it does (and this has been known since the 1980s).

What the REACH students discovered in Spring 2002 (shortly after researchers Mireille Bousquet-Melou and Julian West, working independently of the REACH group, made the same discovery), is that these numbers actually count something. For instance, the number 7, drawn in purple on the front of the shirt, counts what are called the "perfect matchings" of the purple graph shown in the row beneath.

A graph is a set of dots (called vertices) and line segments joining those dots (called edges). The purple graph has 24 vertices and 31 edges. To see what a perfect matching of a graph is and why the purple graph has exacty 7 of them, look at the back of the tee-shirt. A perfect matching of a graph is a way of selecting some of the edges (shown in yellow) so that every vertex belongs to exactly one of the selected edges. All seven of the perfect matchings of the purple graph are shown surrounding the number 7.

You can check for yourself that the red graph shown on the front of the tee-shirt has 2 perfect matchings, and that the green graph has 3. The turquoise graph has 23, which was too many to put on a shirt. It also wasn't feasible to include on the shirt more examples of the "Somos-4 graphs" so that people seeing it would be able to infer what the general rule is for building up larger examples (such as the one with 59 perfect matchings, or the one with 314). At some point soon a preprint will become available, which will give more details about these graphs and how they were discovered. One key tool was a method of "graphical condensation" devised by Eric Kuo (related to Dodgson's condensation algorithm for calculating determinants).

One consequence of this work concerns generalizations of the Somos-4 sequence. For instance, consider the Somos-4 sequence in which the first four terms remain 1,1,1,1 but the recurrence is

                x a(n-1) a(n-3) + y a(n-2)^2
        a(n) = ------------------------------ 
where x and y are indeterminates. The results discovered in Spring 2002 imply that subsequent terms are polynomials in x,y with positive coefficients. (General results proved by Sergey Fomin and Andrei Zelevinsky in their article The Laurentness phenomenon imply that the terms of the sequence are polynomials, but positivity of the coefficients of these "Somos-4 polynomials" had remained conjectural.)

We can also consider the sequence obtained by replacing the initial terms 1,1,1,1 by indeterminates a,b,c,d and computing subsequent terms via either the original Somos-4 recurrence or the version with coefficients x,y. The work of Fomin and Zelevinsky implies that each subsequent term can be written as a a multivariate Laurent polynomial (a generalization of the notion of polynomial, in which variables may be raised to negative powers as well as positive ones). It was observed that the coefficients of these Laurent polynomials were positive, but a rigorous proof only emerged out of the new combinatorial understanding of the Somos-4 sequence in terms of perfect matchings of graphs.

These discoveries were based on some work I did in January 2002, concerning three-dimensional analogues of the Somos-4 recurrence and other recurrence relations. This led me to formulate a number of conjectures, most of which have now been proved. (See the memo A Dozen Laurent Recurrences, which I originally posted on the REACH private web-page in February 2002.)

For more on Somos-4 and other Somos sequences, check out the Somos sequence web-site.

For more on REACH, go to the REACH home-page.

There are no more shirts available for sale, but I'm likely to encourage the REACH students to create a new shirt in Spring 2003. If you send me email, I can add your name to the list of people to notify when other math T-shirts become available in the future.

Last updated by Jim Propp, July 16, 2002.