  We consider properties of the box polynomials, a one variable polynomial
defined over all integer partitions $\lambda$ whose Young diagrams fit in an
$m$ by $n$ box. We show that these polynomials can be expressed by the finite
difference operator applied to the power $x^{m+n}$. Evaluating box polynomials
yields a variety of identities involving set partition enumeration. We extend
the latter identities using restricted growth words and a new operator called
the fast Fourier operator, and consider connections between set partition
enumeration and the chromatic polynomial on graphs. We also give connections
between the box polynomials and the excedance matrix, which encodes
combinatorial data from a noncommutative quotient algebra motivated by the
recurrence for the excedance set statistic on permutations.
