A378626 Table T(n, k) read by upward antidiagonals. T(n,1) = A377137(n), T(n,2) = A377137(A377137(n)), T(n,3) = A377137(A377137(A377137(n))) and so on.

1, 4, 1, 2, 3, 1, 3, 4, 2, 1, 6, 2, 3, 4, 1, 5, 5, 4, 2, 3, 1, 12, 6, 6, 3, 4, 2, 1, 10, 11, 5, 5, 2, 3, 4, 1, 8, 7, 9, 6, 6, 4, 2, 3, 1, 7, 10, 12, 8, 5, 5, 3, 4, 2, 1, 9, 12, 7, 11, 10, 6, 6, 2, 3, 4, 1, 11, 8, 11, 12, 9, 7, 5, 5, 4, 2, 3, 1, 15, 9, 10, 9, 11, 8, 12, 6, 6, 3, 4, 2, 1, 13, 14, 8, 7, 8, 9, 10, 11, 5, 5, 2, 3, 4, 1, 14, 15, 13, 10, 12, 10, 8, 7, 9, 6, 6, 4, 2

Offset: 1

Author

Boris Putievskiy, Dec 02 2024

Keywords

• nonn
• tabl

Comments

The sequence A377137 generates infinite cyclic group under composition. The identity element is A000027.

Each column is 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.

Row n has length A064455(n). The sequence A064455 is non-monotonic.

The array consists of two triangular arrays alternating row by row.

For odd n, row n consists of permutations of the integers from A001844((n-1)/2) to A265225(n-1). For even n, row n consists of permutations of the integers from A130883(n/2) to A265225(n-1).

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

Examples

			Table begins:
  k =      1   2   3   4   5   6
--------------------------------------
  n =  1:  1,  1,  1,  1,  1,  1, ...
  n =  2:  4,  3,  2,  4,  3,  2, ...
  n =  3:  2,  4,  3,  2,  4,  3, ...
  n =  4:  3,  2,  4,  3,  2,  4, ...
  n =  5:  6,  5,  6,  5,  6,  5, ...
  n =  6:  5,  6,  5,  6,  5,  6, ...
  n =  7: 12, 11,  9,  8, 10,  7, ...
  n =  8: 10,  7, 12, 11,  9,  8, ...
  n =  9:  8, 10,  7, 12, 11,  9, ...
  n = 10:  7, 12, 11,  9,  8, 10, ...
  n = 11:  9,  8, 10,  7, 12, 11, ...
  n = 12: 11,  9,  8, 10,  7, 12, ...
  n = 13: 15, 14, 13, 15, 14, 13, ...
  n = 14: 13, 15, 14, 13, 15, 14, ...
  n = 15: 14, 13, 15, 14, 13, 15, ...
Column k = 1 contains the start of A377137. Ord(T(1,1),T(2,1), ... T(15,1)) = 6, ord(T(1,1),T(2,1), ... T(24,1)) = 18, ord(T(1,1),T(2,1), ... T(45,1)) = 90, ord(T(1,1),T(2,1), ... T(112,1)) = 1260, where ord is order of permutation.
The first 6 antidiagonals are:
  1;
  4, 1;
  2, 3, 1;
  3, 4, 2, 1;
  6, 2, 3, 4, 1;
  5, 5, 4, 2, 3, 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, A064455 (row lengths), A265225, A377137, A378127.

Programs

Mathematica

a[n_]:=Module[{L,R,P,Result},L=Ceiling[Max[x/.NSolve[x*(2*(x-1)-Cos[Pi*(x-1)]+5)-4*n==0,x,Reals]]];R=n-If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; P[(L+1)*(2*L-(-1)^L+5)/4]=If[EvenQ[L],3L/2,(L+1)/2];P[3]=2;P=Abs[2*R-If[EvenQ[L],3L/2,(L+1)/2]-If[2*R<=If[EvenQ[L],3L/2,(L+1)/2]+1,2,1]];Res=P+If[EvenQ[L],(L^2-L)/2,(L^2-1)/2];Result=Res;Result] (*A377137*)
composeSequence[a_,n_,k_]:=Nest[a,n,k]
Nmax=15;Kmax=6;T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]

Formula

(T(1,k),T(2,k), ... T(A265225(n),k)) is permutation of the integers from 1 to A265225(n). (T(1,k),T(2,k), ... T(A265225(n),k)) = (T(1,1),T(2,1), ... T(A265225(n),1))^k.