Question: English bell ringers, called change ringers, ring tower and hand bells following a sequence of permutations; however, one of the requirements of change ringing is that a permutation \(p\) may be followed by a permutation \(p^{\prime}\) only if for \(i=1,2, \ldots, n-1\), the number in position \(i\) of \(p\) is in position \(i-1, i\) or \(i+1\) of \(p^{\prime} .\) The number in position 1 of \(p\) may stay in position 1 or move to position 2, and the number in position \(n\) may stay in position \(n\) or move to position \(n-1\) of \(p^{\prime}\). Thus the following sequence of permutations is legal from the perspective of bell ringing: \(\langle 123\rangle,\langle 132\rangle,\langle 312\rangle\), but \(\langle 213\rangle\) is not a legal follow-up to \(\langle 312\rangle\), since number 3 jumped from the front to the back. Find a sequence of all 3 ! permutations on \(\{1,2,3\}\) that is suitable for bell ringing. Do the same for the 4 ! permutations on \(\{1,2,3,4\}\).

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