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

0,4

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.

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

Example

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

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.

Sequence in context: A355342 A130026 A113287 * A096798 A158902 A137587

Adjacent sequences: A195662 A195663 A195664 * A195666 A195667 A195668

Keyword

nonn,tabf

Author

Tilman Piesk, Sep 23 2011

Extensions

Huge edit by Tilman Piesk, Aug 01 2013

Status

approved