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A380815 a(n) = A379343(A378684(n)).

%I A380815 #28 Jun 24 2025 09:29:19
%S A380815 1,5,2,4,3,6,12,9,14,7,11,10,13,8,15,23,20,25,18,27,16,22,21,24,19,26,
%T A380815 17,28,38,35,40,33,42,31,44,29,37,36,39,34,41,32,43,30,45,57,54,59,52,
%U A380815 61,50,63,48,65,46,56,55,58,53,60,51,62,49,64,47,66
%N A380815 a(n) = A379343(A378684(n)).
%C A380815 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 A380815 The sequence can also be regarded as a table read by upward antidiagonals. For n>1, row n joins two consecutive antidiagonals.
%C A380815 The sequence is an intra-block permutation of the positive integers.
%C A380815 Generalization of the Cantor numbering method.
%C A380815 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, A381663 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A376214, A379343, A380200, A378684, A379342, A380245, A381664. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - _Boris Putievskiy_, Apr 09 2025
%H A380815 Boris Putievskiy, <a href="/A380815/b380815.txt">Table of n, a(n) for n = 1..9730</a>
%H A380815 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 A380815 Boris Putievskiy, <a href="/A379343/a379343_1.pdf">The Alternating Group A4: Subgroups and the Cayley Table</a> (2025).
%H A380815 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlternatingGroup.html">Alternating Group</
%H A380815 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A380815 (a(1), a(2), ..., a(A000384(n+1))) is permutation of the positive integers from 1 to A000384(n+1). ord(a(1), a(2), ..., a(A000384(n+1))) = 3, where ord is the order of the permutation.
%F A380815 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 + m 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 if k >= m and k == 0 (mod 2), where m = 2n - 1.
%e A380815 Triangle array begins:
%e A380815   k=    1  2   3  4   5   6   7  8   9
%e A380815   n=1:  1;
%e A380815   n=2:  5, 2,  4, 3,  6;
%e A380815   n=3: 12, 9, 14, 7, 11, 10, 13, 8, 15;
%e A380815 ord(1, 5, 2, ..., 10, 15) = 3.
%e A380815 For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
%e A380815    1,  2,  6,  7, 15, ...
%e A380815    5,  3, 14,  8, 27, ...
%e A380815    4,  9, 13, 18, 26, ...
%e A380815   12, 10, 25, 19, 42, ...
%e A380815   11, 20, 24, 33, 41, ...
%e A380815   ...
%e A380815 Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
%e A380815   1;
%e A380815   4, 1, 3, 2, 5;
%e A380815   6, 3, 8, 1, 5, 4, 7, 2, 9.
%t A380815 T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k<m,If[OddQ[k],k+m,-k+m],If[OddQ[k],k,-k+2*m]]]
%t A380815 Nmax=3;Flatten[Table[T[n,k],{n,1,Nmax},{k,1,4*n-3}]]
%Y A380815 Cf. A000027, A000384, A016813 (row lengths), A376214, A378684, A379342, A379343, A380200, A380245, A380817, A381662, A381663, A381664.
%K A380815 nonn,tabf
%O A380815 1,2
%A A380815 _Boris Putievskiy_, Feb 04 2025
%E A380815 Name corrected by _Pontus von Brömssen_, Jun 24 2025