Abstract
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 [Formula: see text] 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 [Formula: see text] 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.
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