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A361642 Triangle read by rows where row n is a self-inverse permutation of 1..n formed starting from a column 1..n and sliding numbers to the right and down.

%I A361642 #56 Mar 25 2023 13:21:16
%S A361642 1,1,2,1,3,2,1,4,3,2,1,5,3,4,2,1,6,4,3,5,2,1,7,4,3,5,6,2,1,8,5,6,3,4,
%T A361642 7,2,1,9,5,4,3,7,6,8,2,1,10,6,4,8,3,7,5,9,2,1,11,6,8,5,3,9,4,7,10,2,1,
%U A361642 12,7,5,4,10,3,8,9,6,11,2,1,13,7,5,4,6,3,11,10,9,8,12,2,1,14,8,10,11,6,12,3,9,4,5,7,13,2
%N A361642 Triangle read by rows where row n is a self-inverse permutation of 1..n formed starting from a column 1..n and sliding numbers to the right and down.
%C A361642 The transformation process starts with an upwards column of numbers 1..n.
%C A361642 The rightmost number of the topmost row slides right across columns 1 smaller (if any), and then drops down either onto a 2 or more smaller final column if there is one, or otherwise starting a new final column.
%C A361642 Sliding steps continue until reaching a single row (all columns height 1), which is triangle row n.
%C A361642 The final move is always with 2 that slides rightward then one step down to the end of the resulting single row. 1 never moves.
%C A361642 The resulting row is a self-inverse permutation (involution) because the reverse steps are exactly those performed if the row is stood up as the starting column again.
%C A361642 In the case of transforming the column of a composite n, there will be instances when a number sliding sideways will drop down just one step as the changing stack becomes a complete rectangle of rows and columns. Such a number is always greater by 1 than the height of the completed rectangle, whose height and width are divisors of n.
%C A361642 The travel of a sliding number, other than 2, that thus completes a rectangle will continue: it immediately moves sideways again one step, then downwards. Those n's that are prime numbers will not have such number since their changing stack can never form a rectangle with divisors greater than 1, except themselves.
%C A361642 When performing the process in an orthogonal grid where the numbers slide sideways and downward in discrete steps, the total steps for n is an oblong number, the sequence of which is A002378.
%C A361642 In the orthogonal grid, in mid-process, the bounding box of the changing stack of numbers follows the n = x*y curve, and is in exact contact with it at integer x and y points where n is composite.
%H A361642 Thomas Scheuerle, <a href="/A361642/a361642.png">Numbers colored by absolute displacement.</a> Horizontally: row n of the triangle T(n, k). Vertically: k.
%H A361642 Thomas Scheuerle, <a href="/A361642/a361642_1.png">Numbers colored by value.</a> Horizontally: row n of the triangle T(n, k). Vertically: k.
%F A361642 a(floor(((n+3)^2 - 2*n - 3)/2)) = 3, for n > 0. - _Thomas Scheuerle_, Mar 21 2023
%e A361642 Triangle T(n,k) begins:
%e A361642   n/k |  1  2  3  4  5  6  7
%e A361642   ----------------------------
%e A361642   1   |  1;
%e A361642   2   |  1, 2;
%e A361642   3   |  1, 3, 2;
%e A361642   4   |  1, 4, 3, 2;
%e A361642   5   |  1, 5, 3, 4, 2;
%e A361642   6   |  1, 6, 4, 3, 5, 2;
%e A361642   7   |  1, 7, 4, 3, 5, 6, 2;
%e A361642  ...
%e A361642 .
%e A361642 A few snapshots of the process for n = 7, a prime number:
%e A361642 .
%e A361642   7
%e A361642   6
%e A361642   5
%e A361642   4    4
%e A361642   3    3 5    3 5      3
%e A361642   2    2 6    2 6      2 6 5    2 6 5
%e A361642   1    1 7    1 7 4    1 7 4    1 7 4 3    1 7 4 3 5 6 2
%e A361642 .
%e A361642 An example showing some stages of the process for a composite n = 6, with completed rectangles:
%e A361642 .
%e A361642   6
%e A361642   5
%e A361642   4
%e A361642   3    3 4
%e A361642   2    2 5    2 5 3
%e A361642   1    1 6    1 6 4    1 6 4 3 5 2
%e A361642 .
%e A361642 Step-by-step animation frames, showing 8, the rightmost number of the top row, sliding and dropping during its second movement, in the operation for n = 11:
%e A361642 .
%e A361642   4  8         4     8      4        8   4            4            4
%e A361642   3  9  5      3  9  5      3  9  5      3  9  5  8   3  9  5      3  9  5
%e A361642   2 10  7      2 10  7      2 10  7      2 10  7      2 10  7  8   2 10  7
%e A361642   1 11  6      1 11  6      1 11  6      1 11  6      1 11  6      1 11  6  8
%o A361642 (MATLAB)
%o A361642 function a = A361642( max_row )
%o A361642     a = 1;
%o A361642     for r = 2:max_row
%o A361642         p = [1:r];
%o A361642         for k = 2:r-1
%o A361642             j = [1:r];
%o A361642             t1 = find(mod(j,k) == 0);
%o A361642             t2 = find(mod(j,k) ~= 0);
%o A361642             j(t1) = [r:-1:r-length(t1)+1];
%o A361642             j(t2) = [1:length(t2)];
%o A361642             p = p(j);
%o A361642         end
%o A361642         a = [a p];
%o A361642     end
%o A361642 end % _Thomas Scheuerle_, Mar 21 2023
%Y A361642 Cf. A002260, A002378, A000217 (row sums), A361660.
%K A361642 nonn,tabl
%O A361642 1,3
%A A361642 _Tamas Sandor Nagy_, Mar 19 2023