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A383590 a(n) = A378762(A382499(n)).

%I A383590 #11 Jun 08 2025 16:55:54
%S A383590 1,5,3,6,2,4,14,10,12,8,15,7,13,9,11,27,21,25,19,23,17,28,16,26,18,24,
%T A383590 20,22,44,36,42,34,40,32,38,30,45,29,43,31,41,33,39,35,37,65,55,63,53,
%U A383590 61,51,59,49,57,47,66,46,64,48,62,50,60,52,58,54,56
%N A383590 a(n) = A378762(A382499(n)).
%C A383590 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.
%C A383590 The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
%C A383590 The sequence is an intra-block permutation of the positive integers.
%C A383590 Generalization of the Cantor numbering method.
%C A383590 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), A383419 (the inverse permutation), A381968, A381662, A382499, A380817, A382679, A376214, A382680, A378762, A383589, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy link. - _Boris Putievskiy_, Jun 02 2025
%H A383590 Boris Putievskiy, <a href="/A383590/b383590.txt">Table of n, a(n) for n = 1..9730</a>
%H A383590 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 A383590 Boris Putievskiy, <a href="/A378762/a378762.pdf">The Direct Product D4xC2: Subgroups and the Cayley Table</a>.
%H A383590 Groupprops, <a href="https://groupprops.subwiki.org/wiki/Subgroup_structure_of_direct_product_of_D8_and_Z2">Subgroup structure of direct product of D8 and Z2</a>.
%H A383590 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A383590 ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.
%F A383590 T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = 2m - 1 - k if k < m and k == 1 (mod 2), P(n, k) = m + 1 - k if k < m and k == 0 (mod 2), P(n, k) = 3m - 1 - k if k >= m and k == 1 (mod 2), P(n, k) = - m + k if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A383590 Triangle array begins:
%e A383590   k=    1   2   3  4   5  6   7  8   9
%e A383590   n=1:  1;
%e A383590   n=2:  5,  3,  6, 2,  4;
%e A383590   n=3: 14, 10, 12, 8, 15, 7, 13, 9, 11;
%e A383590 (1, 5, 3, ..., 9, 11) = (1, 2, 3, ..., 12, 11) (1, 5, 3, ..., 7, 15). The first permutation on the right-hand side is from Example A378762 and the second from Example A382499.
%e A383590 Ord(1, 5, 3, ... , 9, 11) = 4.
%e A383590 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A383590    1,  3,  4,  8, 11, ...
%e A383590    5,  2, 12,  9, 23, ...
%e A383590    6, 10, 13, 19, 24, ...
%e A383590   14,  7, 25, 18, 40, ...
%e A383590   15, 21, 26, 34, 41, ...
%e A383590   ...
%e A383590 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A383590   1;
%e A383590   4, 2, 5, 1, 3;
%e A383590   8, 4, 6, 2, 9, 1, 7, 3, 5.
%t A383590 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],2m-1-k,m+1-k],If[OddQ[k],3m-1-k,k-m]]]
%t A383590 Nmax=3; Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A383590 Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383722, A383723, A383724.
%K A383590 nonn,tabf
%O A383590 1,2
%A A383590 _Boris Putievskiy_, May 01 2025