%I #13 Sep 13 2024 07:29:45

%S 1,2,1,3,2,3,6,3,2,1,5,5,1,3,5,4,4,4,2,2,1,7,6,8,4,3,5,7,9,7,6,5,4,3,

%T 2,1,8,11,7,11,1,4,5,7,9,10,9,5,7,6,2,4,3,2,1,15,10,9,9,14,6,3,5,7,9,

%U 11,12,8,18,8,8,7,6,4,4,3,2,1,13,12,11,10,12,17,1,6,5,7,9,11,13,14,13,16,6,10,9,8,2,6,5,4,3,2,1

%N Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

%C A208233 presents an algorithm for generating permutations, where each generated permutation is self-inverse.

%C The sequence in each column k possesses two properties: it is both a self-inverse permutation and an intra-block permutation of natural numbers.

%H Boris Putievskiy, <a href="/A375797/b375797.txt">Table of n, a(n) for n = 1..9870</a>

%H 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 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F T(n,k) = P(n,k) + k*(L(n,k)-1)*L(n,k)/2 = P(n,k) + A062707(L(n-1),k), where L(n,k) = ceiling((sqrt(8*n*k+k^2)-k)/(2*k)), R(n,k) = n-k*(L(n,k)-1)*L(n,k)/2, P(n,k) = (((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))+1)*R(n,k)-((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))-1)*(k*L(n,k)+1-R(n,k))/2.

%F T(n,1) = A188568(n). T(1,k) = A093178(k). T(n,n) = A124625(n). L(n,1) = A002024(n). L(n,2) = A000194(n). L(n,3) = A111651(n). L(n,4) = A371355(n). R(n,1) = A002260(n). R(n,2) = A074294(n).

%e Table begins:

%e k= 1 2 3 4 5 6

%e -----------------------------------

%e n= 1: 1, 1, 3, 1, 5, 1, ...

%e n= 2: 2, 2, 2, 3, 2, 5, ...

%e n= 3: 3, 3, 1, 2, 3, 3, ...

%e n= 4: 6, 5, 4, 4, 4, 4, ...

%e n= 5: 5, 4, 8, 5, 1, 2, ...

%e n= 6: 4, 6, 6, 11, 6, 6, ...

%e n= 7: 7, 7, 7, 7, 14, 7, ...

%e n= 8: 9, 11, 5, 9, 8, 17, ...

%e n= 9: 8, 9, 9, 8, 12, 9, ...

%e n= 10: 10, 10, 18, 10, 10, 15, ...

%e n= 11: 15, 8, 11, 6, 11, 11, ...

%e n= 12: 12, 12, 16, 12, 9, 13, ...

%e n= 13: 13, 13, 13, 13, 13, 12, ...

%e n= 14: 14, 19, 14, 23, 7, 14, ...

%e n= 15: 11, 15, 15, 15, 15, 10, ...

%e n= 16: 16, 17, 12, 21, 30, 16, ...

%e n= 17: 20, 16, 17, 17, 17, 8, ...

%e n= 18: 18, 18, 10, 19, 28, 18, ...

%e ... .

%e In column 3, the first 3 blocks have lengths 3,6 and 9. In column 6, the first 2 blocks have lengths 6 and 12. Each block is a permutation of the numbers of its constituents.

%e The first 6 antidiagonals are:

%e 1;

%e 2,1;

%e 3,2,3;

%e 6,3,2,1;

%e 5,5,1,3,5;

%e 4,4,4,2,2,1;

%t T[n_,k_]:=Module[{L,R,P,result},L=Ceiling[(Sqrt[8*n*k+k^2]-k)/(2*k)]; R=n-k*(L-1)*L/2; P=(((-1)^Max[R,k*L+1-R]+1)*R-((-1)^Max[R,k*L+1-R]-1)*(k*L+1-R))/2; result=P+k*(L-1)*L/2]

%t Nmax=18; Table[T[n,k],{n,1,Nmax},{k,1,Nmax}]

%Y Cf. A000194, A002024, A002260, A003991, A062707, A074294, A093178, A111651, A124625, A188568, A208233, A371355.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Aug 29 2024