A227962 Triangle of permutations that assign sona-becs (A227960) to their complements.

0, 1, 0, 3, 1, 2, 0, 7, 3, 5, 1, 6, 2, 4, 0, 15, 7, 12, 3, 13, 5, 9, 1, 14, 6, 10, 11, 2, 4, 8, 0, 31, 15, 26, 7, 28, 12, 20, 3, 29, 13, 22, 23, 5, 9, 17, 1, 30, 14, 24, 25, 6, 27, 10, 11, 18, 19, 2, 21, 4, 8, 16, 0

Offset: 0

Author

Tilman Piesk, Aug 04 2013

Keywords

• nonn
• tabf

Comments

Subgroups of nimber addition (sona, A190939) have complements (defined using their Walsh spectrum). All sona in the same sona-bec (A227960) have complements in a unique sona-bec, which thus can be called its complement.

The permutation in row n of this triangle assigns complementary sona-becs of size 2^n to each other. (It is thus self-inverse.)

Even rows contain fixed points, because some sona-becs with weight 2^(n/2) are their own complements. E.g., in row 4 the fixed points are 3, 5, 10 and 11.

Each row contains the row before as a subsequence.

0 is always complement with A076766(n)-1, so each row ends with 0, and the left column is A076766-1 (not A000225).

Examples

			T(4,1)=7 and T(4,7)=1, so 1 and 7 are complements for n=4.
T(4,3)=3, so 3 is its own complement for n=4.
Triangle begins:
    k = 0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n
0       0
1       1   0
2       3   1   2   0
3       7   3   5   1   6   2   4   0
4      15   7  12   3  13   5   9   1  14   6  10  11   2   4   8   0
		

Links

• Tilman Piesk, Rows 0...7, flattened
• Tilman Piesk, Rows 0...7 (the same with emphasis on subsequences)
• Tilman Piesk, Complement pairs for n=0...7
• Tilman Piesk, Graphic for n=4, complements are symmetric to each other
• Tilman Piesk, Subgroups of nimber addition (Wikiversity)