
312 Algebraic Operads: An Algorithmic Companion
For this weight, the arity is a = 1 + w(n −1) = n
2
, which gives the somewhat
amusing conclusion that the matrix is square if and only if the arity is n
2
.
(Note that this solution is only for a single relation; for two or more relations
we need to multiply the left side of the equation by the number s of relation,
which gives the general solution w = (n + s)/s.)
The second question of invertibility is harder to resolve: it corresponds to
a zero nullspace and a nilpotent operad.
10.3 Nonsymmetric ternary operads
In this section we consider the simplest case of a non-binary operad: a
nonsymmetric operad with ...