A059966 -id:A059966 - cp's OEIS frontend

A342492 Number of compositions of n with weakly increasing first quotients.

1, 1, 2, 4, 7, 11, 17, 26, 37, 52, 73, 95, 125, 163, 208, 261, 330, 407, 498, 607, 734, 881, 1056, 1250, 1480, 1738, 2029, 2359, 2742, 3160, 3635, 4169, 4760, 5414, 6151, 6957, 7861, 8858, 9952, 11148, 12483, 13934, 15526, 17267, 19173, 21252, 23535, 25991

Offset: 0

Gus Wiseman, Mar 16 2021

nonn

Also called log-concave-up compositions.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The composition (4,2,1,2,3) has first quotients (1/2,1/2,2,3/2) so is not counted under a(12), even though the first differences (-2,-1,1,1) are weakly increasing.
The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,2)    (4,1)        (4,2)
                       (2,1,1)    (1,1,3)      (5,1)
                       (1,1,1,1)  (2,1,2)      (1,1,4)
                                  (3,1,1)      (2,1,3)
                                  (1,1,1,2)    (2,2,2)
                                  (2,1,1,1)    (3,1,2)
                                  (1,1,1,1,1)  (4,1,1)
                                               (1,1,1,3)
                                               (2,1,1,2)
                                               (3,1,1,1)
                                               (1,1,1,1,2)
                                               (2,1,1,1,1)
                                               (1,1,1,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The weakly decreasing version is A069916.
The version for differences instead of quotients is A325546.
The strictly increasing version is A342493.
The unordered version is A342497, ranked by A342523.
The strict unordered version is A342516.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A002843 counts compositions with all adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A274199, A318991, A318992, A342527, A342528.

b:= proc(n, q, l) option remember; `if`(n=0, 1, add(
     `if`(q=0 or q>=l/j, b(n-j, l/j, j), 0), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 25 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]
(* Second program: *)
b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[
     If[q == 0 || q >= l/j, b[n - j, l/j, j], 0], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(21)-a(47) from Alois P. Heinz, Mar 25 2021

A342493 Number of compositions of n with strictly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 11, 16, 22, 28, 39, 49, 61, 77, 93, 114, 140, 169, 198, 233, 276, 321, 381, 439, 509, 591, 678, 774, 883, 1007, 1147, 1300, 1465, 1641, 1845, 2068, 2317, 2590, 2881, 3193, 3549, 3928, 4341, 4793, 5282, 5813, 6401, 7027, 7699, 8432, 9221, 10076

Offset: 0

Gus Wiseman, Mar 16 2021

nonn

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The composition (3,1,1,2) has first quotients (1/3,1,2) so is counted under a(7).
The a(1) = 1 through a(7) = 16 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)        (7)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)      (1,6)
              (2,1)  (2,2)    (2,3)    (2,4)      (2,5)
                     (3,1)    (3,2)    (3,3)      (3,4)
                     (1,1,2)  (4,1)    (4,2)      (4,3)
                     (2,1,1)  (1,1,3)  (5,1)      (5,2)
                              (2,1,2)  (1,1,4)    (6,1)
                              (3,1,1)  (2,1,3)    (1,1,5)
                                       (3,1,2)    (2,1,4)
                                       (4,1,1)    (2,2,3)
                                       (2,1,1,2)  (3,1,3)
                                                  (3,2,2)
                                                  (4,1,2)
                                                  (5,1,1)
                                                  (2,1,1,3)
                                                  (3,1,1,2)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A325547.
The weakly increasing version is A342492.
The strictly decreasing version is A342494.
The unordered version is A342498, ranked by A342524.
The strict unordered version is A342517.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A274199 counts compositions with all adjacent parts x < 2y.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528.

b:= proc(n, q, l) option remember; `if`(n=0, 1, add(
     `if`(q=0 or q>l/j, b(n-j, l/j, j), 0), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..55);  # Alois P. Heinz, Mar 25 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]
(* Second program: *)
b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[
     If[q == 0 || q > l/j, b[n - j, l/j, j], 0], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 55] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(21)-a(51) from Alois P. Heinz, Mar 18 2021

A342532 Number of even-length compositions of n with alternating parts distinct.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 9, 14, 28, 44, 83, 136, 250, 424, 757, 1310, 2313, 4018, 7081, 12314, 21650, 37786, 66264, 115802, 202950, 354858, 621525, 1087252, 1903668, 3330882, 5831192, 10204250, 17862232, 31260222, 54716913, 95762576, 167614445, 293356422, 513456686

Offset: 0

Gus Wiseman, Mar 28 2021

nonn

These are finite even-length sequences q of positive integers summing to n such that q(i) != q(i+2) for all possible i.

			The a(2) = 1 through a(7) = 14 compositions:
  (1,1)  (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
         (2,1)  (2,2)  (2,3)  (2,4)      (2,5)
                (3,1)  (3,2)  (3,3)      (3,4)
                       (4,1)  (4,2)      (4,3)
                              (5,1)      (5,2)
                              (1,1,2,2)  (6,1)
                              (1,2,2,1)  (1,1,2,3)
                              (2,1,1,2)  (1,1,3,2)
                              (2,2,1,1)  (1,2,3,1)
                                         (1,3,2,1)
                                         (2,1,1,3)
                                         (2,3,1,1)
                                         (3,1,1,2)
                                         (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..500

The strictly decreasing version appears to be A064428 (odd-length: A001522).
The equal version is A065608 (A342527 with odds).
The weakly decreasing version is A114921 (A342528 with odds).
Including odds gives A224958.
A000726 counts partitions with alternating parts unequal.
A325545 counts compositions with distinct first differences.
A342529 counts compositions with distinct first quotients.
Cf. A000009, A000041, A003242, A008965, A032020, A059966, A062968, A064410, A070211, A106351, A175342, A325546.

qdq[q_]:=And@@Table[q[[i]]!=q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],qdq]],{n,0,15}]

\\ here gf gives A106351 as g.f.
gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}
seq(n)={my(p=gf(n,y)); Vec(sum(k=0, n\2, polcoef(p,k,y)^2))} \\ Andrew Howroyd, Apr 16 2021

G.f.: 1 + Sum_{k>=1} B_k(x)^2 where B_k(x) is the g.f. of column k of A106351. - Andrew Howroyd, Apr 16 2021

Terms a(24) and beyond from Andrew Howroyd, Apr 16 2021

A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 7, 10, 14, 21, 31, 42, 63, 91, 123, 184, 255, 371, 511, 750, 1015, 1519, 2047, 3030, 4092, 6111, 8176, 12222, 16383, 24486, 32767, 49024, 65503, 98175, 131061, 196308, 262143, 392959, 524223, 785910, 1048575, 1572256, 2097151, 3144702, 4194162

Offset: 0

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

Number of Lyndon compositions (aperiodic necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic Lyndon compositions begins:
  (1)  (2)  (3)  (4)    (5)    (6)      (7)
                 (112)  (113)  (114)    (115)
                        (122)  (1122)   (133)
                               (11112)  (223)
                                        (11113)
                                        (11212)
                                        (11122)
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row sums of A179317.

Cf. A008965, A025065, A056476, A059966, A242414, A285037, A317085, A317086, A317087, A318731.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1;
c[n_] := c[n] = If[EvenQ[n], 2^(n/2 - 1) + c[n/2], 2^((n - 1)/2)];
a56503[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])];
a[n_] := DivisorSum[n, MoebiusMu[#] a56503[n/#]&];
Array[a, 45] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)

a(n) = {if(n < 1, n==0, sumdiv(n, d, moebius(d)*(2 + d%2)*(2^(n/d\2)))/(4 - n%2))} \\ Andrew Howroyd, Sep 26 2019

seq(n) = Vec(1 + (1/2)*sum(k=1, n, moebius(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - moebius(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)) + O(x*x^n))) \\ Andrew Howroyd, Sep 27 2019

a(n) = Sum_{d|n} mu(d)*A056503(n/d) for n > 0.
a(n) = Sum_{k=1..2} A285037(n, k). - Andrew Howroyd, Apr 08 2017
G.f.: 1 + (1/2)*Sum_{k>=1} mu(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - mu(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)). - Andrew Howroyd, Sep 27 2019

a(17)-a(45) from Andrew Howroyd, Apr 08 2017

a(0)=1 prepended by Andrew Howroyd, Sep 27 2019

A298971 Number of compositions of n that are proper powers of Lyndon words.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 16, 1, 20, 9, 35, 1, 69, 1, 110, 21, 188, 1, 381, 7, 632, 59, 1184, 1, 2300, 1, 4115, 189, 7712, 25, 14939, 1, 27596, 633, 52517, 1, 101050, 1, 190748, 2247, 364724, 1, 703331, 19, 1342283, 7713, 2581430, 1, 4985609, 193

Offset: 1

Gus Wiseman, Jan 30 2018

nonn

a(n) is the number of compositions of n that are not Lyndon words but are of the form p * p * ... * p where * is concatenation and p is a Lyndon word.

			The a(12) = 16 compositions: 111111111111, 1111211112, 11131113, 112112112, 11221122, 114114, 12121212, 123123, 131313, 132132, 1515, 222222, 2424, 3333, 444, 66.

Cf. A000005, A000031, A000740, A000961, A001045, A008965, A019536, A034691, A051953, A052823, A059966, A060223, A178472, A185700, A296302, A296373.

Table[Sum[DivisorSum[d,MoebiusMu[d/#]*(2^#-1)&]/d,{d,Most@Divisors[n]}],{n,100}]

a(n) = sumdiv(n, d, (2^d-1)*(eulerphi(n/d)-moebius(n/d))/n); \\ Michel Marcus, Jan 31 2018

a(n) = Sum_{d|n} (2^d-1)*(phi(n/d)-mu(n/d))/n.

a(n) = A008965(n) - A059966(n).

A299119 Positive solution to 2^(n-1) = (1/n) * Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 2, 6, 14, 40, 84, 224, 484, 1134, 2480, 5632, 12036, 26624, 56896, 122640, 261078, 557056, 1176876, 2490368, 5237360, 11008704, 23057408, 48234496, 100635144, 209714400, 436154368, 905962860, 1878931264, 3892314112, 8052800160, 16642998272, 34359209436

Offset: 1

Gus Wiseman, Feb 03 2018

nonn

For prime p, a(p) = 2^(p-2)*p. - Jon E. Schoenfield, Feb 03 2018

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000740, A001787, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299149, A299151.

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1, n*2^(n-2)-
       add(a(d)*a(n/d), d=divisors(n) minus {1, n})/2)
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Mar 07 2018

nn=50;
sys=Table[2^(n-1)*n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]

A306669 Number of aperiodic permutation necklaces of weight n.

Original entry on oeis.org

1, 0, 1, 4, 23, 110, 719, 4992, 40302, 362492, 3628799, 39912804, 479001599, 6226974714, 87178289207, 1307673722880, 20922789887999, 355687417744992, 6402373705727999, 121645100223036700, 2432902008176115023, 51090942167993548790, 1124000727777607679999

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

A permutation is aperiodic if every rotation of {1...n} acts on the vertices of the cycle decomposition to produce a different digraph. A permutation necklace is an equivalence class of permutations under the action of rotation of vertices in the cycle decomposition. The corresponding action on words applies m -> m + 1 for m < n and n -> 1, and rotates once to the right. For example, (24531) first becomes (35142) under the application of cyclic rotation, and then is rotated right to give (23514).

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Inequivalent representatives of the a(5) = 23 aperiodic necklace permutations.

Cf. A000031, A000740, A000939, A001037, A059966, A060223, A061417, A086675, A323861, A323865, A323866, A323871.

Cf. A324461, A324512, A324513, A324514.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]]/n,{n,6}]

a(n) = (1/n)*sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

a(n) = A324514(n)/n.

a(n) = (1/n)*Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

A318730 Number of cyclic compositions (necklaces of positive integers) summing to n with adjacent parts (including the last and first part) being indivisible (either way).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 6, 5, 8, 7, 14, 15, 21, 31, 39, 51, 69, 98, 133, 177, 254, 329, 471, 632, 902, 1230, 1710, 2370, 3270, 4591, 6384, 8898, 12429, 17252, 24230, 33783, 47405, 66254, 92860, 130142, 182469, 256262, 359676, 505231, 710059, 997953, 1404215

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(14) = 14 cyclic compositions with adjacent parts indivisible either way:
  (14)
  (3,11) (4,10) (5,9) (6,8)
  (2,5,7) (2,7,5) (3,4,7) (3,7,4)
  (2,3,2,7) (2,3,4,5) (2,5,2,5) (2,5,4,3) (3,4,3,4)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A167606, A285573, A303362, A304713, A316476, A318726, A318727, A318728, A318729, A328601.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And[neckQ[#],And@@Not/@Divisible@@@Partition[#,2,1,1],And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->i%j<>0 && j%i<>0))); vector(n, n, 1 + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

a(n) = A328601(n) + 1. - Andrew Howroyd, Oct 27 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A323872 Number of n X n aperiodic binary toroidal necklaces.

Original entry on oeis.org

1, 2, 2, 54, 4050, 1342170, 1908852102, 11488774559598, 288230375950387200, 29850020237398244599296, 12676506002282260237970435130, 21970710674130840874443091905460038, 154866286100907105149455216472736043777350, 4427744605404865645682169434028029029963535277450

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (Lyndon word) case is A001037.

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Inequivalent representatives of the a(2) = 2 aperiodic necklaces:
  [0 0] [0 1]
  [0 1] [1 1]
Inequivalent representatives of the a(3) = 54 aperiodic necklaces:
  000  000  000  000  000  000  000  000  000
  000  000  001  001  001  001  001  001  001
  001  011  001  010  011  100  101  110  111
.
  000  000  000  000  000  000  000  000  000
  011  011  011  011  011  011  011  111  111
  001  010  011  100  101  110  111  001  011
.
  001  001  001  001  001  001  001  001  001
  001  001  001  001  001  001  010  010  010
  010  011  100  101  110  111  011  101  110
.
  001  001  001  001  001  001  001  001  001
  010  011  011  011  011  011  100  100  100
  111  010  011  101  110  111  011  110  111
.
  001  001  001  001  001  001  001  001  001
  101  101  101  101  110  110  110  110  111
  011  101  110  111  011  101  110  111  011
.
  001  001  001  011  011  011  011  011  011
  111  111  111  011  011  011  101  110  111
  101  110  111  101  110  111  111  111  111

Andrew Howroyd, Table of n, a(n) for n = 0..50

Main diagonal of A323861.
Cf. A000031, A000740, A001037, A027375, A059966, A179043, A184271, A323351.
Cf. A323859, A323860, A323865, A323866, A323871.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],And[apermatQ[#],neckmatQ[#]]&]],{n,4}]

Terms a(5) and beyond from Andrew Howroyd, Aug 21 2019

A334273 Numbers k such that the k-th composition in standard order is both a reversed necklace and a co-necklace.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 45, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 85, 87, 91, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 141, 143, 146, 147

Offset: 1

Gus Wiseman, Apr 25 2020

nonn

A necklace is a finite sequence of positive integers that is lexicographically less than or equal to any cyclic rotation. Co-necklaces are defined similarly, except with greater instead of less.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of all reversed necklace co-necklaces begins:
    0: ()            31: (1,1,1,1,1)       69: (4,2,1)
    1: (1)           32: (6)               71: (4,1,1,1)
    2: (2)           33: (5,1)             73: (3,3,1)
    3: (1,1)         34: (4,2)             74: (3,2,2)
    4: (3)           35: (4,1,1)           75: (3,2,1,1)
    5: (2,1)         36: (3,3)             77: (3,1,2,1)
    7: (1,1,1)       37: (3,2,1)           79: (3,1,1,1,1)
    8: (4)           39: (3,1,1,1)         85: (2,2,2,1)
    9: (3,1)         42: (2,2,2)           87: (2,2,1,1,1)
   10: (2,2)         43: (2,2,1,1)         91: (2,1,2,1,1)
   11: (2,1,1)       45: (2,1,2,1)         95: (2,1,1,1,1,1)
   15: (1,1,1,1)     47: (2,1,1,1,1)      127: (1,1,1,1,1,1,1)
   16: (5)           63: (1,1,1,1,1,1)    128: (8)
   17: (4,1)         64: (7)              129: (7,1)
   18: (3,2)         65: (6,1)            130: (6,2)
   19: (3,1,1)       66: (5,2)            131: (6,1,1)
   21: (2,2,1)       67: (5,1,1)          132: (5,3)
   23: (2,1,1,1)     68: (4,3)            133: (5,2,1)

The aperiodic case is A334266.
Compositions of this type are counted by A334271.
Normal sequences of this type are counted by A334272.
Another ranking of the same compositions is A334274 (binary expansion).
Binary (or reversed binary) necklaces are counted by A000031.
Necklace compositions are counted by A008965.
All of the following pertain to compositions in standard order (A066099):
- Necklaces are A065609.
- Reversed necklaces are A333943.
- Co-necklaces are A333764.
- Reversed co-necklaces are A328595.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed Lyndon words are A334265.
- Reversed co-Lyndon words are A328596.
- Aperiodic compositions are A328594.
Cf. A019536, A034691, A059966, A060223, A329138, A334269, A334270.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[0,100],neckQ[Reverse[stc[#]]]&&coneckQ[stc[#]]&]

cp's OEIS Frontend

A342492 Number of compositions of n with weakly increasing first quotients.

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A342493 Number of compositions of n with strictly increasing first quotients.

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A342532 Number of even-length compositions of n with alternating parts distinct.

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A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.

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A298971 Number of compositions of n that are proper powers of Lyndon words.

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A299119 Positive solution to 2^(n-1) = (1/n) * Sum_{d|n} a(d) * a(n/d).

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A306669 Number of aperiodic permutation necklaces of weight n.

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