A375725 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows, where each row is a permutation of the numbers of its constituents; see Comments.

1, 3, 1, 2, 2, 6, 4, 3, 2, 10, 5, 4, 4, 2, 1, 6, 9, 3, 8, 14, 1, 10, 6, 5, 4, 3, 20, 28, 8, 7, 1, 6, 12, 3, 2, 36, 9, 8, 7, 5, 5, 18, 26, 2, 1, 7, 5, 20, 7, 10, 5, 4, 34, 44, 1

Offset: 1

Author

Boris Putievskiy, Aug 25 2024

Keywords

• nonn
• tabl

Comments

Generalization of the Cantor numbering method for k (k > 1) adjacent diagonals. In this approach, column number k combines k neighboring diagonals. Block number n in column k has length k^2*n - k*(k-1)/2 = A360665(n,k) for n, k > 0.

A208234 presents an algorithm for generating permutations, where each generated permutation is self-inverse.

The sequence in each column k possesses two properties: it is both a self-inverse permutation and an intra-block permutation of natural numbers.

Examples

			Table begins:
  k=      1   2   3   4   5   6
------------------------------------
  n= 1:   1,  1,  6, 10,  1,  1, ...
  n= 2:   3,  2,  2,  2, 14, 20, ...
  n= 3:   2,  3,  4,  8,  3,  3, ...
  n= 4:   4,  4,  3,  4, 12, 18, ...
  n= 5:   5,  9,  5,  6,  5,  5, ...
  n= 6:   6,  6,  1,  5, 10, 16, ...
  n= 7:  10,  7,  7,  7,  7,  7, ...
  n= 8:   8,  8, 20,  3,  8, 14, ...
  n= 9:   9,  5,  9,  9,  9,  9, ...
  n=10:   7, 10, 18,  1,  6, 12, ...
  n=11:  11, 11, 11, 36, 11, 11, ...
  n=12:  14, 20, 16, 12,  4, 10, ...
  n=13:  13, 13, 13, 34, 13, 13, ...
  n=14:  12, 18, 14, 14,  2,  8, ...
  n=15:  15, 15, 15, 32, 15, 15, ...
  n=16:  21, 16, 12, 16, 55,  6, ...
  n=17:  17, 17, 17, 30, 17, 17, ...
  n=18:  19, 14, 10, 18, 53,  4, ...
  n=19:  18, 19, 19, 28, 19, 19, ...
  n=20:  20, 12,  8, 20, 51,  2, ...
  n=21:  16, 21, 21, 26, 21, 21, ...
       ... .
In column 2, the first 3 blocks have lengths 3,7 and 11. In column 3, the first 2 blocks have lengths 6 and 15. In column 6, the first block has a length of 21.
Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
  1;
  3, 1;
  2, 2, 6;
  4, 3, 2, 10;
  5, 4, 4, 2, 1;
  6, 9, 3, 8, 14, 1;
		

Links

• Boris Putievskiy, Table of n, a(n) for n = 1..9870
• Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A208234, A342719, A360665.

Programs

Mathematica

T[n_,k_]:=Module[{L,R,P,result},L=Ceiling[(Sqrt[8*n+1]-1)/(2*k)]; R=n-k*(L-1)*(k(L-1)+1)/2; If[2*R>=k^2*L-k*(k-1)/2+1,P=-Max[R,k^2*L-k*(k-1)/2+1-R]*((-1)^R-1)/2+Min[R,k^2*L-k*(k-1)/2+1-R]*((-1)^R+1)/2,P=Max[R,k^2*L-k*(k-1)/2+1-R]*((-1)^(k^2*L-k*(k-1)/2+1-R)+1)/2-Min[R,k^2*L-k*(k-1)/2+1-R]*((-1)^(k^2*L-k*(k-1)/2+1-R)-1)/2]; result=P+k*(L-1)*(k*(L-1)+1)/2]
Nmax=21; Table[T[n,k],{n,1,Nmax},{k,1,Nmax}]

Formula

T(n,k) = P(n,k) + k*(L(n,k) - 1)*(k*(L(n,k) - 1) + 1)/2 = P(n,k) + A342719(L(n,k) - 1,k)), where L(n,k) = ceiling((sqrt(8*n+1)-1)/(2*k)), R(n,k) = n - k*(L(n,k)-1)*(k*(L(n,k)-1)+1)/2, P(n,k) = - max(R(n,k) , k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^R(n,k) - 1) / 2 + min(R(n,k) , k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^R(n,k) + 1) / 2 if 2R(n,k) ≥k^2*L - k(k-1)/2 + 1, P(n,k) = max(R(n,k) , k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^(k^2*L(n,k) - k(k-1)/2 + 1 - R) + 1) / 2 - min(R, k^2*L(n,k) - k(k-1)/2 + 1 - R) * ((-1)^(k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) - 1) / 2 if 2R < k^2*L(n,k) - k(k-1)/2. + 1.