The maximal Abelian dimension of linear algebras formed by strictly upper triangular matrices

We compute the largest dimension of the Abelian Lie subalgebras contained in the Lie algebra $\mathfrak g_n$ of $n\times n$ strictly upper triangular matrices, where $n\in\mathbb N\setminus\{1\}$. We do this by proving a conjecture, which we previously advanced, about this dimension. We introduce an algorithm and use it first to study the two simplest particular cases and then to study the general case.