An expansion, of the type considered by Hassitt, of the Young operators of S$_{A+1}$ in terms of the two-sided products of the Young operators of S$_{A}$ into the transposition P$_{A,\,A+1}$, is derived in complete generality and explicitly evaluated for the symmetry type of the general Wigner nuclear supermultiplet. The expansion allows of a recursive construction of the Young operators which is simpler than that considered, for example, by Thrall. A complete set of Young operators for S$_{3}$, S$_{4}$, S$_{5}$ and S$_{6}$ is explicitly constructed and tabulated. In an appendix explicit formulae are given for the most general case.