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By using a free monoid of operators on the set of compositions (resp. pairsof compositions), we establish in this paper a bijective correspondence betweenFrobenius standard parabolic (resp. seaweed) subalgebras and certain elementsof this monoid. We prove via this correspondence a conjecture of one of theauthors on the number of Frobenius standard parabolic (resp. seaweed)subalgebras of $\mathrm{sl}_{n}(\Bbbk )$ associated to compositions (resp.pairs of compositions) with $n-t$ parts (resp. parts in total).

On compositions associated to Frobenius parabolic and seaweed subalgebras of $\mathrm{sl}_{n}(\Bbbk )$