A379342 a(n) = A378684(A378684(n)).

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

Offset

1,2

Comments

The 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 Cantor numbering method.

The sequence A378200 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A378200 with itself: A378684(n) = A378200(A378200(n)) = A378200(n)^2, A378762(n) = A378200(n)^3, A379342(n) = A378200(n)^4, A378705(n) = A378200(n)^5. The identity element is A000027(n) = A378200(n)^6. - Boris Putievskiy, Jan 03 2025

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, A378684 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Mar 30 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

Linear sequence: (a(1), a(2), ..., a(A000384(n+1))) is a permutation of the positive integers from 1 to A000384(n+1). ord(a(1), a(2), ..., a(A000384(n+1))) = 3, where ord is the order of the permutation. (a(1), a(2), ..., a(A000384(n+1))) = (A378684(1), A378684(2), ..., A378684(A000384(n+1)))^(-1).

Triangular array T(n,k) for 1 <= k <= 4n - 3 (see Example): T(n,k) = A000384(n-1) + P(n,k), P(n,k) = max(k, 4n - 3 - k) if k == 1 (mod 2), P(n,k) = min(k - 1, 4n - 2 - k) if k == 0 (mod 2).

Example

Triangle array begins:
  k=   1   2   3   4   5   6   7   8   9
  n=1: 1;
  n=2: 5,  2,  4,  3,  6;
  n=3: 14, 7, 12,  9, 11, 10, 13,  8, 15;
  ...
(1, 5, 2, ..., 8, 15) = (A378684(1), A378684(2), A378684(3), ..., A378684(14), A378684(15))^2.
(1, 5, 2, ..., 8, 15) = (A378684(1), A378684(2), A378684(3), ..., A378684(14), A378684(15))^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  2,  6,  9, 15, ...
   5,  3, 12,  8, 23, ...
   4,  7, 13, 18, 26, ...
  14, 10, 25, 19, 40, ...
  11, 16, 24, 31, 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, 1, 3, 2, 5;
  8, 1, 6, 3, 5, 4, 7, 2, 9.

Mathematica

P[n_, k_]:=If[OddQ[k], Max[k, 4 n-3-k], Min[k-1, 4 n-2-k]]
Nmax=3; Flatten[Table[P[n, k]+(n-1)*(2*n-3), {n, 1, Nmax}, {k, 1, 4*n-3}]]

Crossrefs

Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664.

Sequence in context: A249920 A380200 A380815 * A380245 A021660 A266122

Adjacent sequences: A379339 A379340 A379341 * A379343 A379344 A379345

Keyword

nonn,tabf

Author

Boris Putievskiy, Dec 21 2024

Status

approved