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

Author

Boris Putievskiy, Nov 19 2024

Keywords

• nonn
• tabl

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.

Examples

			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;
		

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.

Crossrefs

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

Programs

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}]

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.