OFFSET
1,2
COMMENTS
The permutation that turns a natural ordered into a sequency ordered Walsh matrix of size 2^n is the product of the Gray code permutation A003188(0..2^n-1) and the bit-reversal permutation A030109(n,0..2^n-1).
(This permutation of 2^n elements can be represented by the compression vector [2^(n-1), 3*[2^(n-2)..4,2,1]] with n elements.)
This triangle shows the compression vectors of the unique square roots of these permutations, which correspond to symmetric binary matrices with 2n-1 ones.
(These n X n matrices correspond to graphs that can be described by permutations of n elements, which are shown in A239304.)
Rows of the square array:
T(1,n) = 1,3,6,6,12,12,24,24,48,48,96,96,192,192,384,384,... (compare A003945)
T(2,n) = 1,1,9,18,18,36,36,72,72,144,144,288,288,576,576,... (compare A005010)
Columns of the square array:
T(m,1) = 1,1,5,10,10,20,20,40,40,80,80,160,160,320,320,... (compare A146523)
T(m,2) = 3,1,1,17,34,34,68,68,136,136,272,272,544,544,... (compare A110287)
LINKS
Tilman Piesk, First 140 rows of the triangle, flattened
Tilman Piesk, Sequency ordered Walsh matrix (Wikiversity)
Tilman Piesk, Calculation in MATLAB
EXAMPLE
Triangular array begins:
1
3 1
6 1 5
6 9 1 10
12 18 1 17 10
12 18 33 1 34 20
Square array begins:
1 3 6 6 12 12
1 1 9 18 18 36
5 1 1 33 66 66
10 17 1 1 129 258
10 34 65 1 1 513
20 34 130 257 1 1
The Walsh permutation wp(8,12,6,3) = (0,8,12,4, 6,14,10,2, 3,11,15,7, 5,13,9,1) permutes the natural ordered into the sequency ordered Walsh matrix of size 2^4.
Its square root is wp(6,9,1,10) = (0,6,9,15, 1,7,8,14, 10,12,3,5, 11,13,2,4).
So row 4 of the triangular array is (6,9,1,10).
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tilman Piesk, Mar 14 2014
STATUS
approved