Recently I made a puzzle for my wife that included a cryptogram, which involves an arrangement of the letters A through Z. I used MATLAB (of course) to permute the letters, but in the first arrangement, H was coded as H. I tried again until MATLAB gave me a derangement, a permutation such that none of the letters was in its original position. I later noticed that the letter I was coded as S and S was coded as I. Though I used that permutation, I really wanted one in which no two elements are simply swapped. I called such a permutation an estrangement, and although I later learned of a more technical and mathematical description, I will keep my name.
Write a function to count estrangements—i.e., the permutations of elements in a 1xn vector such that (1) no element is in its original position and (2) no two elements are simply swapped. For example, if the vector is [1 2 3 4], then [3 2 4 1] and [4 1 3 2] would not be allowed under condition 1 and [4 3 2 1] and [2 1 4 3] would not be allowed under condition 2. Return the count as a string.
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