Title: Loop-augmented forests and a variant of the Foulkes' conjecture,
Abstract: A loop-augmented forest is a labeled rooted forest with loops on some of its
roots. By exploiting an interplay between nilpotent partial functions and
labeled rooted forests, we investigate the permutation action of the symmetric
group on loop-augmented forests. Furthermore, we describe an extension of the
Foulkes' conjecture and prove a special case. Among other important outcomes of
our analysis are a complete description of the stabilizer subgroup of an
idempotent in the semigroup of partial transformations and a generalization of
the (Knuth-Sagan) hook length formula.