  We define $P_{n,c}$ to be the set of all diagrams consisting of two rows of
$n$ vertices with edges, each colored with an element in a set of $c$ possible
colors, connecting vertices in different rows. Each vertex can have at most one
edge incident to it, and no edges of the same color can cross. In this paper,
we find a complete set of irreducible representations of $\C P_{n,c}$. We show
that the Bratteli diagram of $\C P_{0,c} \subseteq \C P_{1,c} \subseteq \C
P_{2,c} \subseteq...$ is Pascal's $(c+1)$-simplex, and use this to provide an
alternative proof of the well-known recursive formula for multinomial
coefficients.
