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A382499 Inverse permutation to A381968.

%I A382499 #18 May 30 2025 23:30:53
%S A382499 1,5,3,4,2,6,12,8,14,10,11,9,13,7,15,23,17,25,19,27,21,22,20,24,18,26,
%T A382499 16,28,38,30,40,32,42,34,44,36,37,35,39,33,41,31,43,29,45,57,47,59,49,
%U A382499 61,51,63,53,65,55,56,54,58,52,60,50,62,48,64,46,66
%N A382499 Inverse permutation to A381968.
%C A382499 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 A382499 The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
%C A382499 The sequence is an intra-block permutation of the positive integers.
%C A382499 Generalization of the Cantor numbering method.
%C A382499 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, A381968 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - _Boris Putievskiy_, Apr 17 2025
%C A382499 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 (the inverse permutation), A381662, 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 27 2025
%H A382499 Boris Putievskiy, <a href="/A382499/b382499.txt">Table of n, a(n) for n = 1..9730</a>
%H A382499 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 A382499 Boris Putievskiy, <a href="/A381968/a381968.pdf">The Dihedral Group D4 (I): Subgroups and the Cayley Table</a> (2025 D4 (I)).
%H A382499 Boris Putievskiy, <a href="/A378762/a378762.pdf">The Direct Product D4xC2: Subgroups and the Cayley Table</a> (2025 D4xC2).
%H A382499 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 A382499 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DihedralGroupD4.html">Dihedral Group D_4</a>.
%H A382499 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A382499 ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.
%F A382499 T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = k + m 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 + 2m - 1 if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A382499 Triangle array begins:
%e A382499   k=     1  2   3   4   5  6   7  8   9
%e A382499   n=1:   1;
%e A382499   n=2:   5, 3,  4,  2,  6;
%e A382499   n=3:  12, 8, 14, 10, 11, 9, 13, 7, 15;
%e A382499 (1,5,3,...,7,15)^(-1) = (1,5,3,...,9,15). The permutation on the right-hand side is from Example A381968.
%e A382499 ord(1,5,3,...,7,15) = 4.
%e A382499 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A382499    1,  3,  6, 10, 15, ...
%e A382499    5,  2, 14,  7, 27, ...
%e A382499    4,  8, 13, 19, 26, ...
%e A382499   12,  9, 25, 18, 42, ...
%e A382499   11, 17, 24, 32, 41, ...
%e A382499   ...
%e A382499 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A382499   1;
%e A382499   4, 2, 3, 1, 5;
%e A382499   6, 2, 8, 4, 5, 3, 7, 1, 9.
%t A382499 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],k+m,k],If[OddQ[k],k,-k+2m-1]]]
%t A382499 Nmax=3;Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A382499 Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.
%K A382499 nonn,tabf
%O A382499 1,2
%A A382499 _Boris Putievskiy_, Mar 29 2025