Narrow Results

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 $[x_1,x_2,\dots ,x_m]$ 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.