Solution that is a little less casework. There is symmetry between the mono increasing and mono decreases cases, so just consider the mono increasing case Clearly, the sequence 123456 is somewhat special, so we can just get that out of the way. Consider removing 1 from a sequence. The sequence after must be 23456. If the 1 were first, we would be back to our special sequence. If 1 were second, we would be double counting since we could also remove the 2. If 1 were anywhere else (4 spots left), it would work and be unique (no double counting). The same logic follows when removing 6 from a sequence. So, in total, for sequences that we remove a 1 or a 6 from there are 8 unique sequences and 2 double-counted sequences. Consider removing 2 from a sequence. The sequence after must be 13456. If the 2 were second, we would be back to our special sequence. If 2 were first or third, we would be double counting since we could also remove the 1 or 3, respectively. If 2 were anywhere else (3 spots left), it would work and be unique (no double counting). The same logic follows when removing either 3, 4, or 5 from a sequence. So, in total, for sequences that we remove a 2, 3, 4, or 5 from there are 12 unique sequences and 8 double-counted sequences. That is all the cases. Summing up, we get 1 + 8 + 2/2 + 12 + 8/2 = 26 By symmetry, total answer for mono increasing and decreasing would be 52.
