A379342 - OEIS

%I #17 Jan 15 2025 08:51:06

%S 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,

%T 17,28,44,29,42,31,40,33,38,35,37,36,39,34,41,32,43,30,45,65,46,63,48,

%U 61,50,59,52,57,54,56,55,58,53,60,51,62,49,64,47,66,90,67,88,69,86,71,84,73,82,75,80,77,79,78,81,76,83,74

%N a(n) = A378684(A378684(n)).

%C 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.

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

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

%C Generalization of Cantor numbering method.

%C 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

%H Boris Putievskiy, <a href="/A379342/b379342.txt">Table of n, a(n) for n = 1..9730</a>

%H Boris Putievskiy, <a href="https://arxiv.org/abs/2310.18466">Integer Sequences: Irregular Arrays and Intra-Block Permutations</a>, arXiv:2310.18466 [math.CO], 2023.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F 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).

%F 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 mod 2 = 1, P(n, k) = min(k - 1, 4n - 2 - k) if k mod 2 = 0.

%e Triangle array begins:

%e   k=   1   2   3   4   5   6   7   8   9

%e   n=1: 1;

%e   n=2: 5,  2,  4,  3,  6;

%e   n=3: 14, 7, 12,  9, 11, 10, 13,  8, 15;

%e   ...

%e (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).

%e For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:

%e   1,  2,  6,  9, 15, ...

%e   5,  3, 12,  8, 23, ...

%e   4,  7, 13, 18, 26, ...

%e  14, 10, 25, 19, 40, ...

%e  11, 16, 24, 31, 41, ...

%e   ...

%e Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:

%e   1;

%e   4, 1, 3, 2, 5;

%e   8, 1, 6, 3, 5, 4, 7, 2, 9;

%t P[n_,k_]:=If[OddQ[k],Max[k,4 n-3-k],Min[k-1,4 n-2-k]]

%t Nmax=3;Flatten[Table[P[n,k]+(n-1)*(2*n-3),{n,1,Nmax},{k,1,4 n-3}]]

%Y Cf. A000027, A016813 (row lengths), A000384, A370655, A373498, A374447, A374494, A374531, A375469, A375602, A375725, A378200,  A378684, A378705, A378762.

%K nonn,tabf,changed

%O 1,2

%A _Boris Putievskiy_, Dec 21 2024