  In this paper, by a modification of a previously constructed minimal free
resolution for a transversal monomial ideal, the Betti numbers of this ideal is
explicitly computed. For convenient characteristics of the ground field, up to
a change of coordinates, the ideal of $t$-minors of a generic pluri-circulant
matrix is a transversal monomial ideal . Using a Gr\"obner basis for this
ideal, it is shown that the initial ideal of a generic pluri-circulant matrix
is a stable monomial ideal when the matrix has two square blocks. By means of
the Eliahou-Kervair resolution, the Betti numbers of this initial ideal is
computed and it is proved that, for some significant values of $t$, this ideal
has the same Betti numbers as the corresponding transversal monomial ideal. The
ideals treated in this paper, naturally arise in the study of generic
singularities of algebraic varieties.
