Conjectures and properties: I conjecture that the permutations with both properties, for n >= 4, have the first (n-1)/2 elements greater than or equal to (n+1)/2 for odd n, and the first n/2 elements greater than or equal to (n+2)/2 for even n. For even n, only one permutation appears to have the first element equal to (n+2)/2, following the structure . [(n+2)/2, n, (n+2)/2 + 1, n-1, (n+2)/2 + 2, n-2, . . . . . . ceiling((3*n+2)/4), ceiling((n+2)/4), . . . . . . n/2 - 2, 1 + 2, n/2 - 1, 1 + 1, n/2, 1] . while all the others have the first element strictly greater than (n+2)/2, the last element equal to the first element minus n/2, and the (n/2+1)th element equal to the (n/2)th element minus n/2. The conjecture is the following: - For odd n: suppose we have n = 7, so [1,2,3,4,5,6,7] is the ordered list. Then all the permutations with both properties have the form of [a,b,c,d,e,f,g], with the first (n-1)/2 elements a,b,c >= (n+1)/2 = 4. - For even n: suppose we have n = 8, so [1,2,3,4,5,6,7,8] is the ordered list. Then we have one permutation with the first element equal to (n+2)/2, in this form: [5,8,6,7,3,2,4,1] and all the other permutations in the form of [a,b,c,d,e,f,g,h], with the first n/2 elements a,b,c,d >= 4, with a > 4, and with h = a - 4 and e = d - 4, where for a different value of n, in place of the number 4 we use n/2. As for the other property I have observed for even n, suppose we have a permutation: a) [5, 8, 6, 7, 3, 2, 4, 1] We can obtain another valid one by splitting it in the middle and mirroring each half: [5, 8, 6, 7, | 3, 2, 4, 1] and after mirroring it it becomes: b) [7, 6, 8, 5, | 1, 4, 2, 3] We can obtain a third one by taking the permutation a), splitting it in the middle and substituting the elements as follows: 1 -> n/2 2 -> n/2 - 1 ... n/2 -> 1 n/2 + 1 -> n n/2 + 2 -> n-1 ... n -> n/2 + 1 so in this case the substitutions are: 1->4, 2->3, 3->2, 4->1, 5->8, 6->7, 7->6, 8->5 and the permutation a) becomes: c) [8, 5, 7, 6, | 2, 3, 1, 4] The final permutation can be obtained by splitting and mirroring c), so: d) [6, 7, 5, 8, | 4, 1, 3, 2] This holds for all even n, at least up to n=12. From _Hugo Pfoertner_, Mar 07 2021: (Start) The lexicographically earliest of the a(16) = 256 permutations satisfying the two criteria is [9, 16, 10, 15, 11, 14, 12, 13, 5, 4, 6, 3, 7, 2, 8, 1], the lexicographically latest is [16, 9, 15, 10, 14, 11, 13, 12, 4, 5, 3, 6, 2, 7, 1, 8]. (End) A final property I have observed for even n is that there are only a(n)/4 unique permutations, since all the others can be obtained through substitution, reflection or a combination of both. This holds for at least 4 <= n <= 12.