OFFSET
1,2
COMMENTS
A208233 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)*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.
EXAMPLE
Table begins:
k= 1 2 3 4 5 6
-----------------------------------
n= 1: 1, 1, 3, 1, 5, 1, ...
n= 2: 2, 2, 2, 3, 2, 5, ...
n= 3: 3, 3, 1, 2, 3, 3, ...
n= 4: 6, 5, 4, 4, 4, 4, ...
n= 5: 5, 4, 8, 5, 1, 2, ...
n= 6: 4, 6, 6, 11, 6, 6, ...
n= 7: 7, 7, 7, 7, 14, 7, ...
n= 8: 9, 11, 5, 9, 8, 17, ...
n= 9: 8, 9, 9, 8, 12, 9, ...
n= 10: 10, 10, 18, 10, 10, 15, ...
n= 11: 15, 8, 11, 6, 11, 11, ...
n= 12: 12, 12, 16, 12, 9, 13, ...
n= 13: 13, 13, 13, 13, 13, 12, ...
n= 14: 14, 19, 14, 23, 7, 14, ...
n= 15: 11, 15, 15, 15, 15, 10, ...
n= 16: 16, 17, 12, 21, 30, 16, ...
n= 17: 20, 16, 17, 17, 17, 8, ...
n= 18: 18, 18, 10, 19, 28, 18, ...
... .
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.
The first 6 antidiagonals are:
1;
2,1;
3,2,3;
6,3,2,1;
5,5,1,3,5;
4,4,4,2,2,1;
MATHEMATICA
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]
Nmax=18; Table[T[n, k], {n, 1, Nmax}, {k, 1, Nmax}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Aug 29 2024
STATUS
approved