The front of the tee-shirt shows an example of a combinatorial construct called a "grove". Groves were invented by participants in the R.E.A.C.H. program (Research Experiences in Algebraic Combinatorics at Harvard), both as tools for the study of other problems in algebra and combinatorics, and for their own sake.

The picture shows a grove of order 20, of which there are exactly 3 to the 100th power. The black lines form straight or zig-zag paths (sometimes with some dead-ends thrown in).

You may notice that the paths are straighter and more orderly in corners of the triangle, and more tangled in the middle. The part of the triangle in which the paths are tangled roughly coincides with the interior of the circle inscribed in the triangle, and this is no coincidence; it has been proved that when n is large, nearly every grove of order n has the property that the "tangled" region nearly coincides with the interior of the inscribed circle.

On the other hand, the definition of groves says nothing about circles; the appearance of circles is a hidden consequence of the definition. This in turn is just one special case of a very general phenomenon, in which large combinatorial objects, viewed from a distance, can exhibit large-scale patterns not visible in the definitions.


The front of the tee-shirt shows a random standard grove of order 20, with the primal and dual representations superimposed.

Standard groves were invented/discovered by REACH members as a combinatorial representation for the terms arising from the 4-term "cube recurrence". If one defines F(i,j,k) = x_{i,j,k} for all (i,j,k) in Z^3 with i+j+k in {-1,0,1} and defines F(i,j,k) with i+j+k>1 via the recurrence

           [  F(i-1,j,k)*F(i,j-1,k-1)]   / 
F(i,j,k) = [+ F(i,j-1,k)*F(i-1,j,k-1)]  /  F(i-1,j-1,k-1) 
           [+ F(i,j,k-1)*F(i-1,j-1,k)] /
then each F(i,j,k) can be written as a Laurent polynomial in the x-variables, and the constituent Laurent monomials correspond to the standard groves of order i+j+k.

To see the primal representation of the grove represented by the picture, look at just the line-segments that bisect dark blue diamond-shaped lozenges ("primal edges") and at the endpoints of these line-segments ("primal vertices"). The primal vertices form a triangular grid (not to be confused with the underlying triangular network, only 1/3 of whose corner-points are primal vertices). In graph-theoretic terms, the primal edges form a forest (a graph that contains no cycles). In this forest, every grid-point lies on an edge, and two grid-points v,w on the boundary of the graph are joined by a path of primal edges if and only if v and w are equally distant from one of the three corners of the graph and neither of the other two corners is closer to v or w. Any forest with these properties is a standard grove (and vice versa).

A "dual vertex" lies at the center of each triangle formed by three mutually adjacent primal vertices. In the dual representation of a grove, two dual vertices are joined by dual edge if and only if this line-segment does not cross an edge of the primal graph.

Much of the above description applies only to a special case of groves, namely "standard groves". The way in which the dark and light blue lozenges are incorporated into the figure gives a hint about the way in which the definition of groves needs to be generalized in order to give the full story of the cube recurrence in the presence of arbitrary initial conditions. (In the standard case, the initial conditions were specified by the variables x_{i,j,k} with i+j+k = -1, 0, or 1. Other subsets of Z^3 can play this role, leading to a variety of combinatorial pictures in which the arrangement of the dark and light lozenges is less constrained.) The extra pendant edges around the boundary are shown because their inclusion simplifies the study of general groves.

The number of (standard) groves of order n is 3^{floor(n^2/4)}. There is an efficient iterative algorithm for generating a random grove of order n, in such a way that each of the possible groves of order n has the same likelihood of arising as any other. The figure was generated by this algorithm. One can give a precise definition of what it means to say that the graph is more tangled in the middle, and one can show that as n (the order of the grove) goes to infinity, the tangled subregion of a random grove of order n, rescaled by n to permit comparison between groves of different orders, converges in probability to the interior of a perfect circle.


Gabriel Carroll and David Speyer, "The Cube Recurrence", unpublished manuscript. Gabriel's web-site is

Kyle Petersen and David Speyer, "An Arctic Circle Theorem for Groves", unpublished manuscript. Kyle Petersen's web-site is