Nonvanishing of Abundant-Tree Modules in Free Filippov Algebras

We study the symmetric-group representation on the multilinear component of a free Filippov n-algebra with k brackets. Friedmann, Hanlon, Stanley, and Wachs introduced a submodule spanned by bracketings whose rooted n-ary trees contain an internal vertex all of whose children are internal, and conjectured that this abundant-tree module is nonzero whenever k is greater than n. We prove this conjecture for every n at least 2 and every k greater than n. The smallest abundant tree admits an explicit nonzero evaluation in the standard simple Filippov algebra of dimension n+1. A value-preserving graft then adds one bracket at a time without changing the value at the root, producing a nonzero abundant bracketing for every admissible pair of parameters.