Explicit expressions for Virasoro singular vectors

We present two explicit expressions for generic singular vectors of type $(r,s)$ of the Virasoro algebra. These results follow from the paper of Bauer et al which presented recursive methods to construct the vectors. The expressions presented here generalise the results of Benoit-Saint Aubin for the type $(1,s)$ singular vectors in two different ways: the first simply solves the recursion through the use of partitions; the second gives explicit formulae for the coefficients in a particular expansion. A Mathematica notebook is available which implements the formulae.
Comment: 15 pages; some typos in the text fixed