On the combinatorics of commutators of Lie algebras

Motivated by the combinatorial properties of products in Lie algebras, we investigate the subset of permutations that naturally appears when we write the long commutator $[x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_m]$ as a sum of associative monomials. We characterize this subset and find some useful equivalences. Moreover, we explore properties concerning the action of this subset on sequences of m elements. In particular we describe sequences that share some special symmetries which can be useful in the study of combinatorial properties in graded Lie algebras.