On the algebraic structure of the Lie algebra of vector fields on the line

The author obtains a description of the structure of a representation of the symmetric group $S_n$ in the space of $n$-linear elements of the variety of Lie algebras generated by the Lie algebra of vector fields on the line. It is proved that this space, as an $S_n$-module, is isomorphic to the space of homogeneous harmonic polynomials of degree $n-1$ in $n$ variables.
Bibliography: 16 titles.