OFFSET
1,2
COMMENTS
This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. For n > 0, the length of row n is 4n-3 = A016813(n+1).
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1),a(2), ... a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
LINKS
Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
FORMULA
T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = k if k < m(n) and k mod 2 = 1, P(n, k) = -k + m(n) + 1 if k < m(n) and k mod 2 = 0, P(n, k) = k if k >= m(n) and k mod 2 = 1, P(n, k) = -k + 3 * m(n) - 1 if k >= m(n) and k mod 2 = 0, where m(n) = 2n - 1.
EXAMPLE
Triangle array begins:
k= 1 2 3 4 5 6 7 8 9
n=1: 1;
n=2: 2, 3, 4, 5, 6;
n=3: 7, 10, 9, 8, 11, 14, 13, 12, 15;
(1,2,3... 12,15) = (1,2,3... 12,15)^(-1).
(1,2,3... 12,15) = (1,5,2... 10,15)(1,3,5, ...7,15). The first permutation is from
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
1, 3, 6, 8, 15, ...
2, 5, 9, 12, 20, ...
4, 10, 13, 19, 26, ...
7, 14, 18, 25, 33, ...
11, 21, 24, 34, 41, ...
...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
1,
1, 2, 3, 4, 5;
1, 4, 3, 2, 5, 8, 7, 6, 9;
MATHEMATICA
T[n_, k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1}, If[k<m, If[OddQ[k], k, -k+m+1], If[OddQ[k], k, -k+3*m-1]]]
Nmax=3; Flatten[Table[T[n, k], {n, 1, Nmax}, {k, 1, 4 n-3}]]
CROSSREFS
KEYWORD
nonn,tabf,new
AUTHOR
Boris Putievskiy, Feb 05 2025
STATUS
approved