A381664 a(n) = A380200(A379342(n)).

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

A379343 and A378684 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the alternating group A4. The list of the 12 elements of that group: this sequence, A380245 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A379342, A378684, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Apr 03 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 Alternating Group A4: Subgroups and the Cayley Table (2025).

Eric Weisstein's World of Mathematics, Alternating Group.

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

Formula

ord(a(1), a(2), ..., a(A000384(n+1))) = 3, where ord is the order of the permutation.

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 + m - 1 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:  3,  5, 4,  2,  6;
  n=3: 10, 12, 8, 14, 11, 9, 13, 7, 15;
ord(1, 3, 5, ..., 7, 15) = 3.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,   5,  6, 14, 15, ...
   3,   2,  8,  7, 17, ...
   4,  12, 13, 25, 26, ...
  10,  9,  19, 18, 32, ...
  11, 23,  24, 40, 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;
  2, 4, 3, 1, 5;
  4, 6, 2, 8, 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], m-k, m+k-1], If[OddQ[k], k, -k+2*m-1]]]
Nmax=3; Flatten[Table[T[n, k], {n, 1, Nmax}, {k, 1, 4*n-3}]]

Crossrefs

Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379342, A379343, A380200, A380815, A380817, A381662, A381663, A381664.

Sequence in context: A338843 A379343 A378684 * A381663 A075077 A196771

Adjacent sequences: A381661 A381662 A381663 * A381665 A381666 A381667

Keyword

nonn,tabf

Author

Boris Putievskiy, Mar 03 2025

Extensions

Name corrected by Pontus von Brömssen, Jun 24 2025

Status

approved