  Recently, a new definition for a Wigner distribution function for a
one-dimensional finite quantum system, in which the position and momentum
operators have a finite (multiplicity-free) spectrum, was developed. This
distribution function is defined on discrete phase-space (a finite square
grid), and can thus be referred to as the Wigner matrix. In the current paper,
we compute this Wigner matrix (or rather, the pre-Wigner matrix, which is
related to the Wigner matrix by a simple matrix multiplication) for the case of
the $\mathfrak{su}(2)$ finite oscillator. The first expression for the matrix
elements involves sums over squares of Krawtchouk polynomials, and follows from
standard techniques. We also manage to present a second solution, where the
matrix elements are evaluations of Dyck polynomials. These Dyck polynomials are
defined in terms of the well known Dyck paths. This combinatorial expression of
the pre-Wigner matrix elements turns out to be particularly simple.
