M Theory Lesson 303

Recall that the counting in BCFW recursion hinges on the Catalan and Narayana numbers.

The Narayana numbers may be obtained from the associahedra via its dual simplicial complex, as shown by Fomin and Reading. This Pascal type algorithm computes the so called h-vector of a complex from the face vector. For example, for our favourite associahedron in dimension 3, the face vector is $(1,9,21,14)$, since there are $14$ vertices, $21$ edges and so on. Write these numbers along the diagonal left edge of a Pascal triangle, with ones along the bottom. The h-vector is obtained by taking differences of the entries to the left and below. So the number $8 = 9 - 1$ will be filled in to the right of the original entry $9$. Completing this process until the triangle is filled results in the h-vector on the right hand edge. In this case, we obtain $(1,6,6,1)$, which counts the allowed trees for the 7 gluon scattering amplitudes.

Now the g-theorem fully characterises h-vectors for polytopes. The face vector may be expressed in the form $gM$ (the McMullen correspondence), where $M$ is a certain matrix. In a recent paper it is shown that all minors of $M$ are nonnegative numbers.