  We consider polynomials that are defined as Wronskians of certain sets of
Hermite polynomials. Our main result is a recurrence relation for these
polynomials in terms of those of one or two degrees smaller, which generalizes
the well-known three term recurrence relation for Hermite polynomials. The
polynomials are defined using partitions of natural numbers, and the
coefficients in the recurrence relation can be expressed in terms of the number
of standard Young tableaux of these partitions. Using the recurrence relation,
we provide another recurrence relation and show that the average of the
considered polynomials with respect to the Plancherel measure is very simple.
Furthermore, we show that some existing results in the literature are easy
corollaries of the recurrence relation.
