A376214 a(n) = A379342(A379343(n)).

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 14, 13, 12, 15, 16, 21, 18, 19, 20, 17, 22, 27, 24, 25, 26, 23, 28, 29, 36, 31, 34, 33, 32, 35, 30, 37, 44, 39, 42, 41, 40, 43, 38, 45, 46, 55, 48, 53, 50, 51, 52, 49, 54, 47, 56, 65, 58, 63, 60, 61, 62, 59, 64, 57, 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. For n > 0, the length of row n is 4n-3 = A016813(n-1).

The sequence is a self-inverse permutation of the positive integers.

In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.

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, A000027 (the identity permutation), A381662, A380817, A379343, A380200, A378684, A379342, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Apr 27 2025

A381968 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, A000027 (the identity permutation), A381968, A381662, A382499, A382679, A376217, A382680. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - Boris Putievskiy, Apr 27 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, A381662, A382499, A380817, A382679, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, 383724. 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 Alternating Group A4: Subgroups and the Cayley Table (2025).

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, Alternating Group.

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

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

Formula

a(n) = A380245(A378684(n)).

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n,k) = k 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 + 3m - 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:  2,  3, 4, 5,  6;
  n=3:  7, 10, 9, 8, 11, 14, 13, 12, 15;
(1, 2, 3, ..., 12, 15) = (1, 2, 3, ..., 12, 15)^(-1).
(1, 2, 3, ..., 12, 15) = (1, 5, 2, ..., 10, 15) (1, 3, 5, ..., 7, 15). The first permutation is from Example A380245 and the second from Example A378684.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6,  8, 15, ...
   2,  5,  9, 12, 20, ...
   4, 10, 13, 19, 26, ...
   7, 14, 18, 25, 33, ...
  11, 21, 24, 34, 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,
  1, 2, 3, 4, 5;
  1, 4, 3, 2, 5, 8, 7, 6, 9.

Mathematica

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

Crossrefs

Cf. A000027, A000384, A016813 (row lengths), A056023, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

Sequence in context: A199426 A381839 A119257 * A266645 A376865 A372368

Adjacent sequences: A376211 A376212 A376213 * A376215 A376216 A376217

Keyword

nonn,tabf

Author

Boris Putievskiy, Feb 05 2025

Extensions

Name corrected by Pontus von Brömssen, Jun 24 2025

Status

approved