Generalized Derangements and Anagrams Without Fixed Letters

For a union of disjoint sets a permutation is a generalized derangement if no element is mapped to an element in its own set. Denote the number of such permutations by $P.$ For a given word $A$ denote by $P'$ the number of anagrams of $A$ for which no letter occupies a position which was occupied by the same letter in the original word. In this article we propose several new properties of the very closely related functions $P$ and $P'.$ After some definitions and preliminary observations, we proceed with two recursive algorithms for computing $P$ and $P'$. We use the algorithms to prove several inequalities which allow us to roughly estimate and partially order the values of $P$ and $P'.$ Finally, we turn to the number-theoretical properties of $P'.$ We prove three theorems and propose four corollaries of the last of them. One of the results in this section fully determines when $P'$ is odd. The main approach in the section is splitting the anagrams into classes of equivalence in different ways.