A382499 Inverse permutation to A381968.

1, 5, 3, 4, 2, 6, 12, 8, 14, 10, 11, 9, 13, 7, 15, 23, 17, 25, 19, 27, 21, 22, 20, 24, 18, 26, 16, 28, 38, 30, 40, 32, 42, 34, 44, 36, 37, 35, 39, 33, 41, 31, 43, 29, 45, 57, 47, 59, 49, 61, 51, 63, 53, 65, 55, 56, 54, 58, 52, 60, 50, 62, 48, 64, 46, 66

Offset

1,2

Comments

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.

The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.

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

Generalization of the Cantor numbering method.

A381968 and and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A381968 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - Boris Putievskiy, Apr 17 2025

A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968 (the inverse permutation), A381662, A380817, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy (2025 D4xC2) link. - Boris Putievskiy, May 27 2025

Links

Boris Putievskiy, Table of n, a(n) for n = 1..9730

Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.

Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (2025 D4 (I)).

Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table (2025 D4xC2).

Groupprops, Subgroup structure of direct product of D8 and Z2.

Eric Weisstein's World of Mathematics, Dihedral Group D_4.

Index entries for sequences that are permutations of the natural numbers.

Formula

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = k + m if k < m and k == 1 (mod 2), P(n, k) = k if k < m and k == 0 (mod 2), P(n, k) = k if k >= m and k == 1 (mod 2), P(n, k) = -k + 2m - 1 if k >= m and k == 0 (mod 2), where m = 2n - 1.

Example

Triangle array begins:
  k=     1  2   3   4   5  6   7  8   9
  n=1:   1;
  n=2:   5, 3,  4,  2,  6;
  n=3:  12, 8, 14, 10, 11, 9, 13, 7, 15;
(1,5,3,...,7,15)^(-1) = (1,5,3,...,9,15). The permutation on the right-hand side is from Example A381968.
ord(1,5,3,...,7,15) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6, 10, 15, ...
   5,  2, 14,  7, 27, ...
   4,  8, 13, 19, 26, ...
  12,  9, 25, 18, 42, ...
  11, 17, 24, 32, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  4, 2, 3, 1, 5;
  6, 2, 8, 4, 5, 3, 7, 1, 9.

Mathematica

T[n_, k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1}, If[k<m, If[OddQ[k], k+m, k], If[OddQ[k], k, -k+2m-1]]]
Nmax=3; Flatten[Table[T[n, k], {n, 1, Nmax}, {k, 1, 4*n-3}]]

Crossrefs

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

Sequence in context: A243359 A382854 A222229 * A382680 A381968 A382679

Adjacent sequences: A382496 A382497 A382498 * A382500 A382501 A382502

Keyword

nonn,tabf

Author

Boris Putievskiy, Mar 29 2025

Status

approved