A195665 Consecutive bit-permutations of nonnegative integers.

0, 1, 0, 2, 1, 3, 0, 1, 4, 5, 2, 3, 6, 7, 0, 2, 4, 6, 1, 3, 5, 7, 0, 4, 1, 5, 2, 6, 3, 7, 0, 4, 2, 6, 1, 5, 3, 7, 0, 1, 2, 3, 8, 9, 10, 11, 4, 5, 6, 7, 12, 13, 14, 15, 0, 2, 1, 3, 8, 10, 9, 11, 4, 6, 5, 7, 12, 14, 13, 15, 0, 1, 4, 5, 8, 9, 12

Offset: 0

Author

Tilman Piesk, Sep 23 2011

Keywords

• nonn
• tabf

Comments

All rows of this array are infinite permutations of the nonnegative integers. Row m (counted from 0) is always generated by modifying the sequence of nonnegative integers in the following way: The sequence of integers is written in reverse binary. Than the finite permutation p_m (row m of array A055089) is applied on the digits of all entries.

The rows of the top left n! X 2^n submatrix describe the rotations and reflections of the n-hypercube that preserve the binary digit sums of the vertex numbers. With permutation composition these permutations form the symmetric group S_n.

Applying such a permutation on the binary string of a Boolean function gives the string of a function in the same big equivalence class (compare A227723).

Triangle row m has length 2^n for m in the interval [(n-1)!,n![. The rest of the array row repeats the same pattern. The first digit of the rest is the digit before plus one.

Examples

			Top left corner of array:
0 1 2 3 4 5 6 7
0 2 1 3 4 6 5 7
0 1 4 5 2 3 6 7
0 2 4 6 1 3 5 7
0 4 1 5 2 6 3 7
0 4 2 6 1 5 3 7
The entry in row 2, column 5 (both counted from 0) is 3: 5 in reverse binary is 101, permutation p_2 applied on 101 gives 110, 110 from reverse binary to decimal is 3.
Corresponding rows of the triangle:
0 1
0 2 1 3
0 1 4 5 2 3 6 7
0 2 4 6 1 3 5 7
0 4 1 5 2 6 3 7
0 4 2 6 1 5 3 7
		

Links

• Tilman Piesk, First 120 rows of the triangle, flattened
• Tilman Piesk, 120x32 top left submatrix (human readable)
• Tilman Piesk, 720x64 top left submatrix (computer readable)
• Tilman Piesk, First 720 rows of the triangle
• Tilman Piesk, Bit-permutations (Wikiversity)
• Tilman Piesk, MATLAB code

Crossrefs

The finite permutations in A055089 are applied on the reverse binary digits.

Row 0: A001477.

Row 1: A080412.

Row n!-1 of the triangle is the n-bit bit-reversal permutation. Compare A030109.

Extensions

Huge edit by Tilman Piesk, Aug 01 2013