On compositions associated to Frobenius parabolic and seaweed   subalgebras of $\mathrm{sl}_{n}(\Bbbk )$
  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). 
Michel Duflo and Rupert W.T. Yu
math.RT