OFFSET
1,2
COMMENTS
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.
A208234 presents an algorithm for generating permutations, where each generated permutation is self-inverse.
The sequence in each column k possesses two properties: it is both a self-inverse permutation and an intra-block permutation of natural numbers.
LINKS
Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
FORMULA
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.
EXAMPLE
Table begins:
k= 1 2 3 4 5 6
------------------------------------
n= 1: 1, 1, 6, 10, 1, 1, ...
n= 2: 3, 2, 2, 2, 14, 20, ...
n= 3: 2, 3, 4, 8, 3, 3, ...
n= 4: 4, 4, 3, 4, 12, 18, ...
n= 5: 5, 9, 5, 6, 5, 5, ...
n= 6: 6, 6, 1, 5, 10, 16, ...
n= 7: 10, 7, 7, 7, 7, 7, ...
n= 8: 8, 8, 20, 3, 8, 14, ...
n= 9: 9, 5, 9, 9, 9, 9, ...
n=10: 7, 10, 18, 1, 6, 12, ...
n=11: 11, 11, 11, 36, 11, 11, ...
n=12: 14, 20, 16, 12, 4, 10, ...
n=13: 13, 13, 13, 34, 13, 13, ...
n=14: 12, 18, 14, 14, 2, 8, ...
n=15: 15, 15, 15, 32, 15, 15, ...
n=16: 21, 16, 12, 16, 55, 6, ...
n=17: 17, 17, 17, 30, 17, 17, ...
n=18: 19, 14, 10, 18, 53, 4, ...
n=19: 18, 19, 19, 28, 19, 19, ...
n=20: 20, 12, 8, 20, 51, 2, ...
n=21: 16, 21, 21, 26, 21, 21, ...
... .
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.
Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
1;
3, 1;
2, 2, 6;
4, 3, 2, 10;
5, 4, 4, 2, 1;
6, 9, 3, 8, 14, 1;
MATHEMATICA
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]
Nmax=21; Table[T[n, k], {n, 1, Nmax}, {k, 1, Nmax}]
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Aug 25 2024
STATUS
approved