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A381662 a(n) = A378684(A380200(n)).

%I A381662 #30 Jun 24 2025 09:58:41
%S A381662 1,2,3,4,5,6,9,8,7,10,11,14,13,12,15,20,17,18,19,16,21,22,27,24,25,26,
%T A381662 23,28,35,30,33,32,31,34,29,36,37,44,39,42,41,40,43,38,45,54,47,52,49,
%U A381662 50,51,48,53,46,55,56,65,58,63,60,61,62,59,64,57,66
%N A381662 a(n) = A378684(A380200(n)).
%C A381662 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 A381662 The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
%C A381662 The sequence is a self-inverse permutation of the positive integers.
%C A381662 In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
%C A381662 The sequence is an intra-block permutation of the positive integers.
%C A381662 Generalization of the Cantor numbering method.
%C A381662 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), A380817, A376214, 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 28 2025
%C A381662 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, A382499, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - _Boris Putievskiy_, Apr 28 2025
%C A381662 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, A382499, A380817, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy (2025 D4xC2) link. - _Boris Putievskiy_, May 21 2025
%H A381662 Boris Putievskiy, <a href="/A381662/b381662.txt">Table of n, a(n) for n = 1..9730</a>
%H A381662 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 A381662 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A381662 Boris Putievskiy, <a href="/A381968/a381968.pdf">The Dihedral Group D4 (I): Subgroups and the Cayley Table</a> (2025 D4 (I)).
%H A381662 Boris Putievskiy, <a href="/A378762/a378762.pdf">The Direct Product D4xC2: Subgroups and the Cayley Table</a>(2025 D4xC2).
%H A381662 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 A381662 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%H A381662 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DihedralGroupD4.html">Dihedral Group D_4</a>.
%H A381662 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A381662 a(n) = A379342(A379343(n)) = A381968(A381968(n)).
%F A381662 T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n,k) = - k + m - 1 if k < m and k == 1 (mod 2), P(n,k) = k if k < m and k == 0 (mod 2), P(n,k) = k if k >= m and k == 1 (mod 2), P(n,k) = -k + 3 * m - 1 if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A381662 Triangle array begins:
%e A381662   k=    1  2  3   4   5   6   7   8   9
%e A381662   n=1:  1;
%e A381662   n=2:  2, 3, 4,  5,  6;
%e A381662   n=3:  9, 8, 7, 10, 11, 14, 13, 12, 15;
%e A381662 (1,2,3,...,12,15) = (1,2,3,...,12,15)^(-1).
%e A381662 (1,2,3,...,12,15) = (1,5,2,...,8,15) (1,3,5,...,9,15). The first permutation is from Example A379342 and the second from Example A379343.
%e A381662 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A381662    1,  3,  6, 10, 15, ...
%e A381662    2,  5,  7, 12, 16, ...
%e A381662    4,  8, 13, 19, 26, ...
%e A381662    9, 14, 18, 25, 31, ...
%e A381662   11, 17, 24, 32, 41, ...
%e A381662   ...
%e A381662 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A381662   1;
%e A381662   1, 2, 3, 4, 5;
%e A381662   3, 2, 1, 4, 5, 8, 7, 6, 9.
%t A381662 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],-k+m-1,k],If[OddQ[k],k,-k+3*m-1]]]
%t A381662 Nmax=3;Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A381662 Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.
%K A381662 nonn,tabf
%O A381662 1,2
%A A381662 _Boris Putievskiy_, Mar 03 2025
%E A381662 Name corrected by _Pontus von Brömssen_, Jun 24 2025