A378626 - OEIS

%I #8 Dec 03 2024 12:46:32

%S 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,

%T 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,

%U 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

%N 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.

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

%C 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.

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

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

%C 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).

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

%H Boris Putievskiy, <a href="/A378626/b378626.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(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.

%e Table begins:

%e   k =      1   2   3   4   5   6

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

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

%e   n =  2:  4,  3,  2,  4,  3,  2, ...

%e   n =  3:  2,  4,  3,  2,  4,  3, ...

%e   n =  4:  3,  2,  4,  3,  2,  4, ...

%e   n =  5:  6,  5,  6,  5,  6,  5, ...

%e   n =  6:  5,  6,  5,  6,  5,  6, ...

%e   n =  7: 12, 11,  9,  8, 10,  7, ...

%e   n =  8: 10,  7, 12, 11,  9,  8, ...

%e   n =  9:  8, 10,  7, 12, 11,  9, ...

%e   n = 10:  7, 12, 11,  9,  8, 10, ...

%e   n = 11:  9,  8, 10,  7, 12, 11, ...

%e   n = 12: 11,  9,  8, 10,  7, 12, ...

%e   n = 13: 15, 14, 13, 15, 14, 13, ...

%e   n = 14: 13, 15, 14, 13, 15, 14, ...

%e   n = 15: 14, 13, 15, 14, 13, 15, ...

%e 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.

%e The first 6 antidiagonals are:

%e   1;

%e   4, 1;

%e   2, 3, 1;

%e   3, 4, 2, 1;

%e   6, 2, 3, 4, 1;

%e   5, 5, 4, 2, 3, 1;

%t 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*)

%t composeSequence[a_,n_,k_]:=Nest[a,n,k]

%t Nmax=15;Kmax=6;T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]

%Y Cf. A000027, A064455 (row lengths), A265225, A377137, A378127.

%K nonn,tabl,new

%O 1,2

%A _Boris Putievskiy_, Dec 02 2024