M Theory Lesson 85

It would be interesting to look at Zagier's conjecture

$d_n = d_{n - 2} + d_{n - 3}$

using associahedra. After all, the relations between MZV values come from the decomposition of an associahedron face as a product of lower dimensional associahedra. For example, the square faces of the 14 vertex K4 polytope arise as products of the K2 intervals, which represent a basic associator. Thus 1-operad combinatorics gives a way to count MZV relations. The dimension of an MZV space of weight $n$ should be related to the difference between the total number of possible arguments (the ordered partitions of n) and the conditions imposed by the combinatorics.

Euler considered a generating function for the partition function, namely $P^{-1}$ for

$P(x) = \prod_{1}^{\infty} (1 - x^n)$

For $n = 3$ we have $P_3 = 3$ since 3 may be written as 1+1+1 or 1+2 or 3. Subtracting 2 kinds of relation for the K4 faces leads us to suspect that $d_3$ is in fact one. That was very rough, but it is nice to think about how generating functions for MZV spaces relate to generating functions from operads.