A227960 -id:A227960 - cp's OEIS frontend

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

Tilman Piesk, Aug 04 2013

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).

			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

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)

A227963 Small equivalence classes (A227722) of subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 5, 6, 15, 17, 18, 51, 20, 85, 105, 24, 102, 90, 60, 255, 257, 258, 771, 260, 1285, 1545, 264, 1542, 1290, 780, 3855, 272, 4369, 4641, 5185, 6273, 288, 4626, 4386, 6210, 5250, 816, 13107, 15555, 320, 5140, 6180, 4420, 4740, 1360, 21845

Offset: 0

Tilman Piesk, Aug 08 2013

nonn

Each entry of this sequence represents the same small equivalence class (sec) of Boolean functions as the corresponding entry of A190939. While A190939 represents each sec by the unique odd number among the numeric values of its functions, this sequence represents each sec by the smallest among these numbers (as an entry of A227722).
All big equivalence classes (bec) of Boolean functions are also small equivalence classes. So all entries in the sequence of sona-becs (A227960) are also in this sequence of sona-secs.
This sequence takes its order from A190939, so it is not monotonic. Thus it is not a subsequence of A227722, and does not contain A227960 as a subsequence.
First entries: 1, 3, 5, 6, 15, 17, 18, 51, 20, 85, 105, 24, 102, 90, 60, 255.
First entries in numerical order: 1, 3, 5, 6, 15, 17, 18, 20, 24, 51, 60, 85, 90, 102, 105, 255.

			A190939(3) = 9. 9 belongs to the sec A227722(4) = 6. So a(3) = 6.
A190939(8) = 65. 65 belongs to the sec A227722(10) = 20. So a(8) = 20.

Tilman Piesk, Table of n, a(n) for n = 0..2824
Tilman Piesk, Table of n, A190939(n), a(n) for n = 0..2824 and the same with a graphical explanation for n = 0..66
Tilman Piesk, Subgroups of nimber addition (Wikiversity)
Tilman Piesk, Small equivalence classes of Boolean functions

cp's OEIS Frontend

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

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