On compositions associated to Frobenius parabolic and seaweed subalgebras of ${\mathrm{s}\mathrm{l}}_n(\mathbb{k})$

By using a free monoid of operators on the set of compositions (resp. pairs of compositions), we establish in this paper a bijective correspondence between Frobenius standard parabolic (resp. seaweed) subalgebras and certain elements of this monoid. We prove via this correspondence a conjecture of one of the authors on the number of Frobenius standard parabolic (resp. seaweed) subalgebras of ${\mathrm{s}\mathrm{l}}_n(\mathbb{k})$ associated to compositions (resp. pairs of compositions) with $n-t$ parts (resp. parts in total).