Loop-augmented forests and a variant of the Foulkes' conjecture

TL;DR

This paper introduces loop-augmented forests, explores their symmetry properties, extends Foulkes' conjecture, and provides new algebraic and combinatorial results including stabilizer subgroups and a hook length formula.

Contribution

It presents a novel combinatorial structure called loop-augmented forests, extends Foulkes' conjecture, and derives new algebraic and combinatorial formulas and descriptions.

Findings

Complete description of stabilizer subgroup of an idempotent

Extension of Foulkes' conjecture with a proven special case

Generalization of the hook length formula

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.