A375376 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.

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, 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, 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

Offset: 1

Author

Pontus von Brömssen, Aug 14 2024

Keywords

• nonn
• tabl

Comments

Each row is a permutation of the positive integers.

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

Examples

			Array begins:
   n=1: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=2: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=3: 1, 2, 3, 5, 4, 7,  6,  8, 11,  9, 12, 10, 15, 16, 13, ...
   n=4: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=5: 1, 2, 3, 4, 5, 7,  6,  8,  9, 11, 15, 10, 12, 16, 13, ...
   n=6: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=7: 1, 2, 3, 4, 7, 5,  6, 10, 13,  8,  9, 11, 14, 12, 15, ...
   n=8: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=9: 1, 3, 2, 7, 4, 5,  8,  6, 15,  9, 11, 16, 10, 12, 17, ...
  n=10: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
  n=11: 1, 2, 4, 3, 7, 5, 13,  6,  8, 10, 14,  9, 11, 22, 16, ...
  n=12: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
  n=13: 1, 2, 4, 3, 5, 7, 13,  6, 10,  8, 14,  9, 15, 11, 12, ...
  n=14: 1, 2, 3, 4, 5, 7,  6, 10,  8,  9, 11, 12, 13, 14, 15, ...
  n=15: 1, 2, 3, 5, 4, 9,  6,  7, 13, 21,  8, 10, 17, 11, 14, ...
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:
   k | power tower | colex rank T(7,k)
   --+-------------+------------------
   1 |     2 = 2   |        1
   2 |     3 = 3   |        2
   3 |     4 = 4   |        3
   4 |   2^2 = 4   |        4
   5 |   2^3 = 8   |        7
   6 |   3^2 = 9   |        5
   7 |   4^2 = 16  |        6
   8 |   2^4 = 16  |       10
   9 | 2^2^2 = 16  |       13
  10 |   3^3 = 27  |        8
  11 |   4^3 = 64  |        9
  12 |   3^4 = 81  |       11
  13 | 3^2^2 = 81  |       14
  14 |   4^4 = 256 |       12
  15 | 4^2^2 = 256 |       15
		

Links

• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

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