A239304 Triangle of permutations corresponding to the compressed square roots of Gray code * bit-reversal permutation (A239303).

1, 1, 2, 3, 1, 2, 4, 2, 1, 3, 2, 5, 4, 1, 3, 2, 5, 6, 3, 1, 4, 6, 2, 3, 7, 5, 1, 4, 7, 3, 2, 6, 8, 4, 1, 5, 3, 8, 7, 2, 4, 9, 6, 1, 5, 3, 8, 9, 4, 2, 7, 10, 5, 1, 6, 9, 3, 4, 10, 8, 2, 5, 11, 7, 1, 6, 10, 4, 3, 9

Offset: 1

Author

Tilman Piesk, Mar 14 2014

Keywords

• nonn
• tabl

Comments

The symmetrical binary matrices corresponding to the rows of A239303 can be interpreted as adjacency matrices of undirected graphs. These graphs are chains where one end is connected to itself, so they can be interpreted as permutations. The end connected to itself is always the first element of the permutation, i.e., on the left side of the triangle.

Columns of the square array:

T(m,1) = A008619(m) = 1,2,2,3,3...

T(m,2) = 1,1,1...

T(m,3) = A028242(m+3) = 3,2,4,3,5,4,6,5,7,6,8,7,9,8,10,9,11,10,12...

T(m,4) = m+3 = 4,5,6...

T(m,5) = A084964(m+4) = 2,5,3,6,4,7,5,8,6,9,7,10,8,11,9,12,10,13...

T(m,6) = 2,2,2...

T(m,7) = A168230(m+5) = 6,3,7,4,8,5,9,6,10,7,11,8,12,9,13,10,14...

T(m,8) = m+6 = 7,8,9...

T(m,9) = A152832(m+9) = 3,8,4,9,5,10,6,11,7,12,8,13,9,14,10,15...

T(m,10) = 3,3,3...

Diagonals of the square array:

T(n,n) = a(A001844(n)) = 1,1,4,7,4,2,9,14,7,3,14,21,10,4,19,28,13,5,24...

T(n,2n-1) = a(A064225(n)) = 1,2,3...

T(2n-1,n) = a(A081267(n)) = 1,1,5,10,6,2,12,21,11,3,19,32,16,4,26,43,21...

Examples

			Triangular array begins:
  1
  1 2
  3 1 2
  4 2 1 3
  2 5 4 1 3
  2 5 6 3 1 4
Square array begins:
  1 1 3 4 2 2
  2 1 2 5 5 2
  2 1 4 6 3 2
  3 1 3 7 6 2
  3 1 5 8 4 2
  4 1 4 9 7 2
Row 5 of A239303 is the vector (12,18,1,17,10), which corresponds to the following binary matrix:
  0 0 1 1 0
  0 1 0 0 1
  1 0 0 0 0
  1 0 0 0 1
  0 1 0 1 0
Interpreted as an adjacency matrix it describes the following graph, where each number is connected to its neighbors, and only the 2 is connected to itself:
  2 5 4 1 3
This is row 5 of the triangular array.
		

Links

• Tilman Piesk, First 140 rows of the triangle, flattened
• Tilman Piesk, Sequency ordered Walsh matrix (Wikiversity)
• Tilman Piesk, Calculation in MATLAB

Crossrefs

Cf. A239303, A028242, A084964, A168230, A152832.