Inspired by this question.
Let's consider the 11-letter word "MATHEMATICS". The general question is to find how many words it is possible to have if we can make 4 permutations of letters.
Each letter has a position in the word, thus when I say 1 permutation, we exchange the position of 2 letters. For example, with 1 permutation it is possible to have "AMTHEMATICS", but not "SMATHEMATIC". How much word is it possible to make with 4 permutations, if ...
- a.1) we want exactly 4 permutations and it is forbidden to have permutation $p$ and its inverse permutation. But it is possible to permute an "M" with another "M".
- a.2) same as a.1) but it is forbidden to permute "M" with the other "M".
Let's say that by permutations, we mean changing the letters in the word in such a way that 4 permutations could lead to "TICSMATHEMA" (when a letter enter a new position, the other letters are pushed). How much word is it possible to make with 4 permutations, if we want exactly 4 permutations, that means 4 letters changing positions?
It seems to me that there is no trivial solution. At least I don't see it. Of course, it would be interesting to have a general solution for the case of a word with $n$ letters (possibly not all different) with $k$ permutations.