  In this paper, we study a family of generating functions whose coefficients
are polynomials that enumerate partitions in lower order ideals of Young's
lattice. Our main result is that this family satisfies a rational recursion and
are therefore rational functions. As an application, we calculate the
asymptotic behavior of the cardinality of lower order ideals for the ``average"
partition of fixed length and give a homological interpretation of this result
in relation to Grassmannians and their Schubert varieties.
