A378198 - OEIS

%I #9 Dec 03 2024 12:46:07

%S 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,

%T 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,

%U 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

%N 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.

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

%C 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).

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

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

%C Generalization of the Cantor numbering method.

%H Boris Putievskiy, <a href="/A378198/b378198.txt">Table of n, a(n) for n = 1..9870</a>

%H Boris Putievskiy, <a href="https://arxiv.org/abs/2310.18466">Integer Sequences: Irregular Arrays and Intra-Block Permutations</a>, arXiv:2310.18466 [math.CO], 2023.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F (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.

%e Table begins:

%e   k =      1   2   3   4   5   6

%e --------------------------------------

%e   n =  1:  1,  1,  1,  1,  1,  1, ...

%e   n =  2:  2,  2,  2,  2,  2,  2, ...

%e   n =  3:  4,  3,  4,  3,  4,  3, ...

%e   n =  4:  3,  4,  3,  4,  3,  4, ...

%e   n =  5:  5,  5,  5,  5,  5,  5, ...

%e   n =  6:  6,  6,  6,  6,  6,  6, ...

%e   n =  7:  7,  7,  7,  7,  7,  7, ...

%e   n =  8: 10, 16,  9, 13, 14, 17, ...

%e   n =  9: 13, 14, 17, 12, 11,  8, ...

%e   n = 10: 16,  9, 13, 14, 17, 12, ...

%e     ...

%e Column k = 1 contains the start of A375602.

%e 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.

%e The first 6 antidiagonals are:

%e   1;

%e   2, 1;

%e   4, 2, 1;

%e   3, 3, 2, 1;

%e   5, 4, 4, 2, 1;

%e   6, 5, 3, 3, 2, 1;

%t 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]

%t Nmax=10; Kmax=6; T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]

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

%K nonn,tabl,new

%O 1,2

%A _Boris Putievskiy_, Nov 19 2024