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.