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

%I A376214 #54 Jun 24 2025 09:58:29
%S A376214 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,
%T A376214 23,28,29,36,31,34,33,32,35,30,37,44,39,42,41,40,43,38,45,46,55,48,53,
%U A376214 50,51,52,49,54,47,56,65,58,63,60,61,62,59,64,57,66
%N A376214 a(n) = A379342(A379343(n)).
%C A376214 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).
%C A376214 The sequence is a self-inverse permutation of the positive integers.
%C A376214 In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
%C A376214 The sequence is an intra-block permutation of the positive integers.
%C A376214 Generalization of the Cantor numbering method.
%C A376214 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
%C A376214 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
%C A376214 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
%H A376214 Boris Putievskiy, <a href="/A376214/b376214.txt">Table of n, a(n) for n = 1..9730</a>
%H A376214 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 A376214 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A376214 Boris Putievskiy, <a href="/A381968/a381968.pdf">The Dihedral Group D4 (I): Subgroups and the Cayley Table</a> (2025 D4 (I)).
%H A376214 Boris Putievskiy, <a href="/A378762/a378762.pdf">The Direct Product D4xC2: Subgroups and the Cayley Table</a> (2025 D4xC2).
%H A376214 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 A376214 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</a>.
%H A376214 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DihedralGroupD4.html">Dihedral Group D_4</a>.
%H A376214 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A376214 a(n) = A380245(A378684(n)).
%F A376214 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.
%e A376214 Triangle array begins:
%e A376214   k=    1   2  3  4   5   6   7   8   9
%e A376214   n=1:  1;
%e A376214   n=2:  2,  3, 4, 5,  6;
%e A376214   n=3:  7, 10, 9, 8, 11, 14, 13, 12, 15;
%e A376214 (1, 2, 3, ..., 12, 15) = (1, 2, 3, ..., 12, 15)^(-1).
%e A376214 (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.
%e A376214 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A376214    1,  3,  6,  8, 15, ...
%e A376214    2,  5,  9, 12, 20, ...
%e A376214    4, 10, 13, 19, 26, ...
%e A376214    7, 14, 18, 25, 33, ...
%e A376214   11, 21, 24, 34, 41, ...
%e A376214   ...
%e A376214 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A376214   1,
%e A376214   1, 2, 3, 4, 5;
%e A376214   1, 4, 3, 2, 5, 8, 7, 6, 9.
%t A376214 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]]]
%t A376214 Nmax=3;Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A376214 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.
%K A376214 nonn,tabf
%O A376214 1,2
%A A376214 _Boris Putievskiy_, Feb 05 2025
%E A376214 Name corrected by _Pontus von Brömssen_, Jun 24 2025