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

1,2

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.

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.

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.

Example

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).

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

Crossrefs

Cf. A064455, A209278, A265225, A377137.

Sequence in context: A108127 A207376 A213197 * A049277 A214917 A260316

Adjacent sequences: A378124 A378125 A378126 * A378128 A378129 A378130

Keyword

nonn,tabf

Author

Boris Putievskiy, Nov 17 2024

Status

approved