A378198 Table T(n, k) read by upward antidiagonals. T(n,1) = A375602(n), T(n,2) = A375602(A375602(n)), T(n,3) = A375602(A375602(A375602(n))) and so on.

1, 2, 1, 4, 2, 1, 3, 3, 2, 1, 5, 4, 4, 2, 1, 6, 5, 3, 3, 2, 1, 7, 6, 5, 4, 4, 2, 1, 10, 7, 6, 5, 3, 3, 2, 1, 13, 16, 7, 6, 5, 4, 4, 2, 1, 16, 14, 9, 7, 6, 5, 3, 3, 2, 1, 8, 9, 17, 13, 7, 6, 5, 4, 4, 2, 1, 11, 10, 13, 12, 14, 7, 6, 5, 3, 3, 2, 1, 14, 8, 16, 14, 11, 17, 7, 6, 5, 4, 4, 2, 1, 17, 17, 10, 9, 17, 8, 12, 7, 6, 5, 3, 3, 2, 1, 19, 12, 12, 16, 13, 12, 10, 11, 7

Offset

1,2

Comments

Sequence A375602 generates an infinite cyclic group under composition. The identity element is A000027.

Each column is triangle read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n has length n(n^2 + 1)/2 = A006003(n).

Each column is an intra-block permutation of the positive integers.

For n > 1, each row combines n consecutive antidiagonals.

Generalization of the Cantor numbering method.

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.

Index entries for sequences that are permutations of the natural numbers.

Formula

(T(1,k),T(2,k), ... T(A002817(n),k)) is permutation of the integers from 1 to A002817(n). (T(1,k),T(2,k), ... T(A002817(n),k)) = (T(1,1),T(2,1), ... T(A002817(n),1))^k.

Example

Table begins:
  k =      1   2   3   4   5   6
--------------------------------------
  n =  1:  1,  1,  1,  1,  1,  1, ...
  n =  2:  2,  2,  2,  2,  2,  2, ...
  n =  3:  4,  3,  4,  3,  4,  3, ...
  n =  4:  3,  4,  3,  4,  3,  4, ...
  n =  5:  5,  5,  5,  5,  5,  5, ...
  n =  6:  6,  6,  6,  6,  6,  6, ...
  n =  7:  7,  7,  7,  7,  7,  7, ...
  n =  8: 10, 16,  9, 13, 14, 17, ...
  n =  9: 13, 14, 17, 12, 11,  8, ...
  n = 10: 16,  9, 13, 14, 17, 12, ...
    ...
Column k = 1 contains the start of A375602.
Ord(T(1,1),T(2,1), ... T(7,1)) = 2, ord(T(1,1),T(2,1), ... T(21,1)) = 18, ord(T(1,1),T(2,1), ... T(55,1)) = 1980, ord(T(1,1),T(2,1), ... T(120,1)) = 51480, where ord is order of permutation.
The first 6 antidiagonals are:
  1;
  2, 1;
  4, 2, 1;
  3, 3, 2, 1;
  5, 4, 4, 2, 1;
  6, 5, 3, 3, 2, 1;

Mathematica

a[n_]:=Module[{L, Ld, Rd, P, Result}, L=Ceiling[(Sqrt[4*Sqrt[8*n+1]-3]-1)/2]; Ld=Ceiling[(Sqrt[8*n+1]-1)/2]; Rd=n-(Ld-1)*Ld/2; P=L*Rd+Ld-L*(L+1)/2-Max[Rd-(L^2-L+2)/2, 0]*(Max[Rd-(L^2-L+2)/2, 0]+1)/2; Result=P+(L-1)*L*(L^2-L+2)/8; Result] (*A375602*) composeSequence[a_, n_, k_]:=Nest[a, n, k]
Nmax=10; Kmax=6; T=Table[composeSequence[a, n, k], {n, 1, Nmax}, {k, 1, Kmax}]

Crossrefs

Cf. A000027, A006003 (row lengths), A002817, A370655, A373498, A374447, A374494, A374531, A375602, A375725.

Sequence in context: A088423 A006839 A268267 * A205395 A367647 A243070

Adjacent sequences: A378195 A378196 A378197 * A378199 A378200 A378201

Keyword

nonn,tabl,new

Author

Boris Putievskiy, Nov 19 2024

Status

approved