A374447 Inverse permutation to A373498.

2, 1, 3, 9, 4, 7, 6, 8, 5, 10, 20, 11, 18, 13, 16, 15, 17, 12, 19, 14, 21, 35, 22, 33, 24, 31, 26, 29, 28, 30, 23, 32, 25, 34, 27, 36, 54, 37, 52, 39, 50, 41, 48, 43, 46, 45, 47, 38, 49, 40, 51, 42, 53, 44, 55

Offset: 1

Author

Boris Putievskiy, Jul 08 2024

Keywords

• nonn
• tabf

Comments

Triangle read by rows where row n is a block of length 4*n-1 which is a permutation of the numbers of its constituents.

Generalization of the Cantor numbering method for two adjacent diagonals. A pair of neighboring diagonals are combined into one block.

The sequence is an intra-block permutation of positive integers.

The sequence A373498 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A373498 with itself: A374494 = A373498(A373498) = A373498^2, A370655 = A373498^3, A374531 = A373498^4, A374447 = A373498^5. The identity element is A000027 = A373498^6. - Boris Putievskiy, Aug 03 2024

Examples

			Triangle begins:
     k = 1   2   3   4   5   6   7   8   9  10  11
  n=1:   2,  1,  3;
  n=2:   9,  4,  7,  6,  8,  5, 10;
  n=3:  20, 11, 18, 13, 16, 15, 17, 12, 19, 14, 21;
The triangle's rows can be arranged as two successive upward antidiagonals in an array:
    2,  3,  7, 10, 16, 21, ...
    1,  4,  5, 13, 14, 26, ...
    9,  8, 18, 19, 31, 34, ...
    6, 11, 12, 24, 25, 41, ...
   20, 17, 33, 32, 50, 51, ...
   15, 22, 23, 39, 40. 60, ...
Subtracting (n-1)*(2*n-1) from each term in row n is a permutation of 1 .. 4*n-1:
    2,1,3,
    6,1,4,3,5,2,7,
   10,1,8,3,6,5,7,2,9,4,11
   ...
The inverse permutation of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^(-1) = (2,1,3),
   (2,6,4,3,5,1,7)^(-1) = (6,1,4,3,5,2,7),
   (2,8,4,10,6,5,7,3,9,1,11)^(-1) = (10,1,8,3,6,5,7,2,9,4,11).
The 5th power of each permutation in example A373498 is equal to the corresponding permutation above:
   (2,1,3)^5 = (2,1,3),
   (2,6,4,3,5,1,7)^5 = (6,1,4,3,5,2,7),
   (2,8,4,10,6,5,7,3,9,1,11)^5 = (10,1,8,3,6,5,7,2,9,4,11).
		

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. A000027, A004767 (row lengths), A204164, A370655, A373498, A374494, A374531.

Programs

Mathematica

Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1); P=Which[R<=2*L&&Mod[R,2]==1,4*L-R-1,R<=2*L&&Mod[R,2]==0,R-1,R>2*L&&Mod[R,2]==1,R,R>2*L&&Mod[R,2]==0,-2*L+R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}]
Nmax=21;
a[n_]:=Module[{L,R,P,Result},L=Ceiling[(Sqrt[8*n+1]-1)/4];
R=n-(L-1)*(2*L-1); P=Which[R<=2*L-1&&Mod[R,2]==1,R+1,R<=2*L-1&&Mod[R,2]==0,R+2*L,R>2*L-1&&Mod[R,2]==1,R,R>2*L-1&&Mod[R,2]==0,4*L-1-R];
Result=P+(L-1)*(2*L-1);
Result]
Table[a[n],{n,1,Nmax}] (* A373498 *)
Table[a[a[a[a[a[n]]]]],{n,1,Nmax}] (* this sequence *)

Formula

Linear sequence:

a(n) = P(n) + (L(n)-1)*(2*L(n)-1), where L(n) = ceiling((sqrt(8*n+1)-1)/4),

L(n) = A204164(n), P(n) = 4*L(n) - R(n) - 1, if R(n) <= 2*L(n) and R(n) mod 2 = 1, P(n) = R(n) - 1, if R(n) <= 2*L(n) and R(n) mod 2 = 0, P(n) = R(n), if R(n) > 2*L(n) and R(n) mod 2 = 1, P(n) = - 2*L(n) + R(n), if R(n) > 2*L(n) and R(n) mod 2 = 0.

a(n) = A373498(A373498(A373498(A373498(A373498(n))))).

Triangular array T(n,k) for 1 <= k <= 4*n-1 (see Example):

T(n,k) = (n-1)*(2*n-1) + P(n,k), where P(n,k) = 4*n - k - 1, if k <= 2*n and k mod 2 = 1, P(n,k) = k-1, if k <= 2*n and k mod 2 = 0, P(n,k) = k, if k > 2*n and k mod 2 = 1, P(n,k) = -2*n + k, if k > 2*n and k mod 2 = 0.