  Let $k[X] = k[x_{i,j}: i = 1,..., m; j = 1,..., n]$ be the polynomial ring in
$m n$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote
by $I_2(X)$ the ideal generated by the $2 \times 2$ minors of the generic $m
\times n$ matrix $[x_{i,j}]$. We give a closed formulation for the dimensions
of the $k$-vector space $k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q))$ as $q$
varies over all positive integers, i.e., we give a closed form for the
generalized Hilbert-Kunz function of the determinantal ring $k[X]/I_{2}[X]$. We
also give a closed formulation of dimensions of related quotients of
$k[X]/I_{2}[X]$. In the process we establish a formula for the numbers of some
compositions (ordered partitions of integers), and we give a proof of a new
binomial identity.
