  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 $\mathrm{sl}_{n}(\Bbbk )$ associated to compositions (resp.
pairs of compositions) with $n-t$ parts (resp. parts in total).
