A378127 Inverse permutation to A377137.

1, 3, 4, 2, 6, 5, 10, 9, 11, 8, 12, 7, 14, 15, 13, 20, 21, 19, 22, 18, 23, 17, 24, 16, 27, 26, 28, 25, 35, 34, 36, 33, 37, 32, 38, 31, 39, 30, 40, 29, 43, 44, 42, 45, 41, 53, 54, 52, 55, 51, 56, 50, 57, 49, 58, 48, 59, 47, 60, 46, 64, 63, 65, 62, 66, 61, 76, 75, 77, 74, 78, 73, 79, 72, 80, 71, 81, 70, 82, 69, 83, 68, 84, 67, 88, 89, 87, 90, 86, 91

Offset: 1

Author

Boris Putievskiy, Nov 17 2024

Keywords

• nonn
• tabf

Comments

Array read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block.

Row n has length A064455(n). The sequence A064455 is non-monotonic.

Subtracting (n^2 - n)/2 if n is even from each term in row n produces a permutation of 1 .. 3n/2. Subtracting (n^2 - 1)/2 if n is odd from each term in row n produces a permutation of 1 .. (n+1)/2.

These permutations are inverses of the corresponding permutations from A377137. The algorithm used to generate them is described in A209278.

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

Examples

			Array begins:
     k =  1   2   3   4   5   6
  n=1:    1;
  n=2:    3,  4,  2;
  n=3:    6,  5;
  n=4:   10,  9, 11,  8, 12, 7;
 The triangular arrays alternate by row: n=1 and n=3 comprise one, and n=2 and n=4 comprise the other. Subtracting 1, 4, and 6 from the elements of rows 2, 3, and 4, respectively, produces permutations:
  1;
  2, 3, 1;
  2, 1;
  4, 3, 5, 2, 6, 1;
  ...
These permutations are the inverses of those in Example A377137, listed in the same order.
(2,3,1)^(-1) = (3,1,2); (2,1)^(-1) = (2,1); (4,3,5,2,6,1)^(-1) = (6,4,2,1,3,5).
		

Links

• Boris Putievskiy, Table of n, a(n) for n = 1..9940
• 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. A064455, A209278, A265225, A377137.

Programs

Mathematica

b[n_]:=(4n+1+(2n-1)*(-1)^n)/4;P[n_,k_]:=If[EvenQ[b[n]-k],(b[n]-k+2)/2,(b[n]+k+1)/2];Res[n_,k_]:=P[n,k]+(-(-1)^n*n+(-1)^n+2 n^2-n-1)/4;
Nmax=4;resultTable=Table[Res[n,k],{n,1,Nmax},{k,1,b[n]}]//Flatten

Formula

Array T(n,k) (see Example):

T(n, k) = P(n, k) + A265225(n-1), where

P(n, k) = (b(n) - k + 2)/2 if mod(b(n) - k, 2) = 0,

P(n, k) = (b(n) + k + 1)/2 if mod(b(n) - k, 2) = 1.

b(n) = (4n + 1 + (2n - 1) * (-1)^n)/4 is the length of the row n.