A375376 - OEIS

%I #4 Aug 17 2024 13:57:27

%S 1,2,1,3,2,1,4,3,2,1,5,4,3,2,1,6,5,5,3,2,1,7,6,4,4,3,2,1,8,7,7,5,4,3,

%T 2,1,9,8,6,6,5,4,3,2,1,10,9,8,7,7,5,4,3,3,1,11,10,11,8,6,6,7,4,2,2,1,

%U 12,11,9,9,8,7,5,5,7,3,2,1,13,12,12,10,9,8,6,6,4,4,4,2,1

%N Square array read by antidiagonals: Let n = Sum_{i=1..m} 2^e_i be the binary expansion of n, let S be the set {e_i+2; 1 <= i <= m}, and let X be the sequence of power towers built of numbers in S, sorted first by their height and then colexicographically. The n-th row of the array gives the permutation of indices which reorders X by magnitude. In case of ties, keep the colexicographic order.

%C Each row is a permutation of the positive integers.

%C If n is a power of 2, the set S contains a single number and the n-th row is the identity permutation.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%e Array begins:

%e    n=1: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...

%e    n=2: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...

%e    n=3: 1, 2, 3, 5, 4, 7,  6,  8, 11,  9, 12, 10, 15, 16, 13, ...

%e    n=4: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...

%e    n=5: 1, 2, 3, 4, 5, 7,  6,  8,  9, 11, 15, 10, 12, 16, 13, ...

%e    n=6: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...

%e    n=7: 1, 2, 3, 4, 7, 5,  6, 10, 13,  8,  9, 11, 14, 12, 15, ...

%e    n=8: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...

%e    n=9: 1, 3, 2, 7, 4, 5,  8,  6, 15,  9, 11, 16, 10, 12, 17, ...

%e   n=10: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...

%e   n=11: 1, 2, 4, 3, 7, 5, 13,  6,  8, 10, 14,  9, 11, 22, 16, ...

%e   n=12: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...

%e   n=13: 1, 2, 4, 3, 5, 7, 13,  6, 10,  8, 14,  9, 15, 11, 12, ...

%e   n=14: 1, 2, 3, 4, 5, 7,  6, 10,  8,  9, 11, 12, 13, 14, 15, ...

%e   n=15: 1, 2, 3, 5, 4, 9,  6,  7, 13, 21,  8, 10, 17, 11, 14, ...

%e For n = 7 = 2^0 + 2^1 + 2^2, the set S is {0+2, 1+2, 2+2} = {2, 3, 4}. The smallest power towers formed by 2's, 3's, and 4's, together with their colex ranks are:

%e    k | power tower | colex rank T(7,k)

%e    --+-------------+------------------

%e    1 |     2 = 2   |        1

%e    2 |     3 = 3   |        2

%e    3 |     4 = 4   |        3

%e    4 |   2^2 = 4   |        4

%e    5 |   2^3 = 8   |        7

%e    6 |   3^2 = 9   |        5

%e    7 |   4^2 = 16  |        6

%e    8 |   2^4 = 16  |       10

%e    9 | 2^2^2 = 16  |       13

%e   10 |   3^3 = 27  |        8

%e   11 |   4^3 = 64  |        9

%e   12 |   3^4 = 81  |       11

%e   13 | 3^2^2 = 81  |       14

%e   14 |   4^4 = 256 |       12

%e   15 | 4^2^2 = 256 |       15

%Y Cf. A185969, A299229, A375374 (3rd row), A375377 (the inverse permutation to each row).

%K nonn,tabl

%O 1,2

%A _Pontus von Brömssen_, Aug 14 2024