A375725 - OEIS

%I #11 Sep 13 2024 07:29:16

%S 1,3,1,2,2,6,4,3,2,10,5,4,4,2,1,6,9,3,8,14,1,10,6,5,4,3,20,28,8,7,1,6,

%T 12,3,2,36,9,8,7,5,5,18,26,2,1,7,5,20,7,10,5,4,34,44,1

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

%C Generalization of the Cantor numbering method for k (k > 1) adjacent diagonals. In this approach, column number k combines k neighboring diagonals. Block number n in column k has length k^2*n - k*(k-1)/2 = A360665(n,k) for n, k > 0.	

%C A208234 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="/A375725/b375725.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)*(k*(L(n,k) - 1) + 1)/2 = P(n,k) + A342719(L(n,k) - 1,k)), where L(n,k) = ceiling((sqrt(8*n+1)-1)/(2*k)),  R(n,k) = n - k*(L(n,k)-1)*(k*(L(n,k)-1)+1)/2, P(n,k) = - max(R(n,k) , k^2*L(n,k)  - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^R(n,k)  - 1) / 2 + min(R(n,k) , k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^R(n,k)  + 1) / 2  if 2R(n,k)  ≥k^2*L - k(k-1)/2 + 1, P(n,k)  = max(R(n,k) , k^2*L(n,k)  - k(k-1)/2 + 1 - R(n,k) ) * ((-1)^(k^2*L(n,k)  - k(k-1)/2 + 1 - R) + 1) / 2 - min(R, k^2*L(n,k) - k(k-1)/2 + 1 - R) * ((-1)^(k^2*L(n,k) - k(k-1)/2 + 1 - R(n,k) ) - 1) / 2  if  2R <  k^2*L(n,k)  - k(k-1)/2. + 1.

%e Table begins:

%e   k=      1   2   3   4   5   6

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

%e   n= 1:   1,  1,  6, 10,  1,  1, ...

%e   n= 2:   3,  2,  2,  2, 14, 20, ...

%e   n= 3:   2,  3,  4,  8,  3,  3, ...

%e   n= 4:   4,  4,  3,  4, 12, 18, ...

%e   n= 5:   5,  9,  5,  6,  5,  5, ...

%e   n= 6:   6,  6,  1,  5, 10, 16, ...

%e   n= 7:  10,  7,  7,  7,  7,  7, ...

%e   n= 8:   8,  8, 20,  3,  8, 14, ...

%e   n= 9:   9,  5,  9,  9,  9,  9, ...

%e   n=10:   7, 10, 18,  1,  6, 12, ...

%e   n=11:  11, 11, 11, 36, 11, 11, ...

%e   n=12:  14, 20, 16, 12,  4, 10, ...

%e   n=13:  13, 13, 13, 34, 13, 13, ...

%e   n=14:  12, 18, 14, 14,  2,  8, ...

%e   n=15:  15, 15, 15, 32, 15, 15, ...

%e   n=16:  21, 16, 12, 16, 55,  6, ...

%e   n=17:  17, 17, 17, 30, 17, 17, ...

%e   n=18:  19, 14, 10, 18, 53,  4, ...

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

%e   n=20:  20, 12,  8, 20, 51,  2, ...

%e   n=21:  16, 21, 21, 26, 21, 21, ...

%e        ... .

%e In column 2, the first 3 blocks have lengths 3,7 and 11. In column 3, the first 2 blocks have lengths 6 and 15. In column 6, the first block has a length of 21.

%e Each block is a permutation of the numbers of its constituents.

%e The first 6 antidiagonals are:

%e   1;

%e   3, 1;

%e   2, 2, 6;

%e   4, 3, 2, 10;

%e   5, 4, 4, 2, 1;

%e   6, 9, 3, 8, 14, 1;

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

%t Nmax=21; Table[T[n,k],{n,1,Nmax},{k,1,Nmax}]

%Y Cf. A208234, A342719, A360665.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Aug 25 2024