A permutation \(\phi \colon \{ 1,2, \ldots, n \} \to \{ 1,2, \ldots, n \}\) is called a well-mixed if \(\phi (\{1,2, \ldots, k \}) \neq \{1,2, \ldots, k \}\) for each \(k<n\). What is the number of well-mixed permutations of \(\{ 1,2, \ldots, 15 \}\)?