  Chessboard complexes and their relatives have been one of important recurring
themes of topological combinatorics. Closely related ``cycle-free chessboard
complexes'' have been recently introduced by Ault and Fiedorowicz as a tool for
computing symmetric analogues of the cyclic homology of algebras. We study
connectivity properties of these complexes and prove a result that confirms a
strengthened conjecture of Ault and Fiedorowicz.
