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A382680 a(n) = A382499(A380817(n)).

%I A382680 #18 Jun 08 2025 21:42:35
%S A382680 1,5,3,4,2,6,12,10,14,8,11,7,13,9,15,23,21,25,19,27,17,22,16,24,18,26,
%T A382680 20,28,38,36,40,34,42,32,44,30,37,29,39,31,41,33,43,35,45,57,55,59,53,
%U A382680 61,51,63,49,65,47,56,46,58,48,60,50,62,52,64,54,66
%N A382680 a(n) = A382499(A380817(n)).
%C A382680 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 A382680 The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
%C A382680 The sequence is a self-inverse permutation of the positive integers.
%C A382680 In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
%C A382680 The sequence is an intra-block permutation of the positive integers.
%C A382680 Generalization of the Cantor numbering method.
%C A382680 A381968 and  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, A382679 (the inverse permutation), A000027 (the identity permutation), A381968, A381662, A382499, A380817, A376214. For subgroups and the Cayley table of the group D4 see Putievskiy (D4 (I)) link. - _Boris Putievskiy_, Apr 27 2025
%C A382680 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, A376214, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - _Boris Putievskiy_, Jun 02 2025
%H A382680 Boris Putievskiy, <a href="/A382680/b382680.txt">Table of n, a(n) for n = 1..9730</a>
%H A382680 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 A382680 Boris Putievskiy, <a href="/A381968/a381968.pdf">The Dihedral Group D4 (I): Subgroups and the Cayley Table</a>.
%H A382680 Boris Putievskiy, <a href="/A378762/a378762.pdf">The Direct Product D4xC2: Subgroups and the Cayley Table</a>.
%H A382680 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 A382680 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DihedralGroupD4.html">Dihedral Group D_4</a>.
%H A382680 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A382680 T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = m + k if k < m and k == 1 (mod 2), P(n, k) = m - k + 1 if k < m and k == 0 (mod 2), P(n, k) = 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 A382680 Triangle array begins:
%e A382680   k=    1   2   3  4   5  6   7  8   9
%e A382680   n=1:  1;
%e A382680   n=2:  5,  3,  4, 2,  6;
%e A382680   n=3: 12, 10, 14, 8, 11, 7, 13, 9, 15;
%e A382680 (1,5,3,...,9,15) = (1,5,3,...,9,15)^(-1).
%e A382680 (1,5,3,...,9,15) = (1,5,3,...,7,15) (1,2,3,...,10,7,...,14,15). The first permutation on the right-hand side is from Example A382499 and the second from Example A380817.
%e A382680 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A382680     1,  3,  6,  8, 15, ...
%e A382680     5,  2, 14,  9, 27, ...
%e A382680     4, 10, 13, 19, 26, ...
%e A382680    12,  7, 25, 18, 42, ...
%e A382680    11, 21, 24, 34, 41, ...
%e A382680    ...
%e A382680 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A382680   1,
%e A382680   4, 2, 3, 1, 5,
%e A382680   6, 4, 8, 2, 5, 1, 7, 3, 9.
%t A382680 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],m+k,m-k+1],If[OddQ[k],k,k-m]]]
%t A382680 Nmax= 3;Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A382680 Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A383419, A383589, A383590, A383722, A383723, A383724.
%K A382680 nonn,tabf
%O A382680 1,2
%A A382680 _Boris Putievskiy_, Apr 03 2025