On compositions associated to Frobenius parabolic and seaweed subalgebras of \(\mathrm{sl}_{n}(\Bbbk)\) (Q2791854)

Let \(\mathbf k\) be an algebraically closed field of characteristic zero. Let, further, \(F_{n,p}\) denote the number of Frobenius standard parabolic subalgebras in \(sl_n(\mathbf k)\) which correspond to compositions of \(n\) with \(p\) parts. Finally, let \(\tilde{F}_{n,p}\) denote the number of Frobenius standard seaweed subalgebras in \(sl_n(\mathbf k)\) which correspond to pairs of partitions of \(n\) having together \(p\) parts. The main result of the paper under review says that, for large \(n\), the numbers \(F_{2n,n+1-t}\), \(F_{2n+1,n+1-t}\) and \(\tilde{F}_{n,n+1-t}\), where \(t\in\mathbb{N}\), are polynomials in \(n\). This establishes a conjecture, originally formulated by the first author. A principal idea of the proof is to establish a bijection between subalgebras in question and certain elements of a free monoid of operators on the set of compositions.