Classification of commutative subalgebras of length $n-2$ in the algebra of $n\times n$ matrices over algebraically closed fields

In this paper, commutative subalgebras of length $n-2$ in the algebra of matrices of order $n$ over algebraically closed fields are classified up to similarity. In other terms, commutative algebras having length one less than the maximum value are described. It is shown that for an arbitrary fixed order of matrices, the number of pairwise non-conjugate algebras of the indicated type is finite. Using the number of partitions of natural numbers, a formula is obtained for the number of different algebras as a function of the order of the matrices.