In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete $k$-uniform hypergraph. We show that the coloring complex of a complete $k$-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the $(n-k-1)^{ st}$ homology group of the cyclic coloring complex of a complete $k$-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose $r$-faces consist of all ordered set partitions $[B_1, \hdots, B_{ r+2}]$ where none of the $B_i$ contain a hyperedge of the complete $k$-uniform hypergraph $H$ and where $1 \in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of $\C[x_1, \hdots, x_n]/ \{ x_{ i_1} \hdots x_{ i_k} \mid i_{ 1} \hdots i_{ k}$ is a hyperedge of $H \}$.
Categories: math.CO