A281013 -id:A281013 - cp's OEIS frontend

A059966 a(n) = (1/n) * Sum_{ d divides n } mu(n/d) * (2^d - 1).

1, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182, 4080, 7710, 14532, 27594, 52377, 99858, 190557, 364722, 698870, 1342176, 2580795, 4971008, 9586395, 18512790, 35790267, 69273666, 134215680, 260300986, 505286415, 981706806

Offset: 1

Roland Bacher, Mar 05 2001

Dimensions of the homogeneous parts of the free Lie algebra with one generator in 1,2,3, etc. (Lie analog of the partition numbers).
This sequence is the Lie analog of the partition sequence (which gives the dimensions of the homogeneous polynomials with one generator in each degree) or similarly, of the partitions into distinct (or odd numbers) (which gives the dimensions of the homogeneous parts of the exterior algebra with one generator in each dimension).
The number of cycles of length n of rectangle shapes in the process of repeatedly cutting a square off the end of the rectangle. For example, the one cycle of length 1 is the golden rectangle. - David Pasino (davepasino(AT)yahoo.com), Jan 29 2009
In music, the number of distinct rhythms, at a given tempo, produced by a continuous repetition of measures with identical patterns of 1's and 0's (where 0 means no beat, and 1 means one beat), where each measure allows for n possible beats of uniform character, and when counted under these two conditions: (i) the starting and ending times for the measure are unknown or irrelevant and (ii) identical rhythms that can be produced by using a measure with fewer than n possible beats are excluded from the count. - Richard R. Forberg, Apr 22 2013
Richard R. Forberg's comment does not hold for n=1 because a(1)=1 but there are the two possible rhythms: "0" and "1". - Herbert Kociemba, Oct 24 2016
The comment does hold for n=1 as the rhythm "0" can be produced by using a measure of 0 beats and so is properly excluded from a(1)=1 by condition (ii) of the comment. - Travis Scott, May 28 2022
a(n) is also the number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n. - Gus Wiseman, Dec 19 2017
Mobius transform of A008965. - Jianing Song, Nov 13 2021
a(n) is the number of cycles of length n for the map x->1 - abs(2*x-1) applied on rationals 0Michel Marcus, Jul 16 2025

			a(4)=3: the 3 elements [a,c], [a[a,b]] and d form a basis of all homogeneous elements of degree 4 in the free Lie algebra with generators a of degree 1, b of degree 2, c of degree 3 and d of degree 4.
From _Gus Wiseman_, Dec 19 2017: (Start)
The sequence of Lyndon compositions organized by sum begins:
  (1),
  (2),
  (3),(12),
  (4),(13),(112),
  (5),(14),(23),(113),(122),(1112),
  (6),(15),(24),(114),(132),(123),(1113),(1122),(11112),
  (7),(16),(25),(115),(34),(142),(124),(1114),(133),(223),(1213),(1132),(1123),(11113),(1222),(11212),(11122),(111112). (End)

C. Reutenauer, Free Lie algebras, Clarendon press, Oxford (1993).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Nicolas Andrews, Lucas Gagnon, Félix Gélinas, Eric Schlums, and Mike Zabrocki, When are Hopf algebras determined by integer sequences?, arXiv:2505.06941 [math.CO], 2025. See p. 17.
S. V. Duzhin and D. V. Pasechnik, Groups acting on necklaces and sandpile groups, Journal of Mathematical Sciences, August 2014, Volume 200, Issue 6, pp 690-697. See page 85. - N. J. A. Sloane, Jun 30 2014
Seok-Jin Kang and Myung-Hwan Kim, Free Lie Algebras, Generalized Witt Formula and the Denominator Identity, Journal of Algebra 183, 560-594 (1996).
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Jakob Oesinghaus, Quasi-symmetric functions and the Chow ring of the stack of expanded pairs, arXiv:1806.10700 [math.AG], 2018.
Robert Schneider, Andrew V. Sills, and Hunter Waldron, On the q-factorization of power series, arXiv:2501.18744 [math.CO], 2025. See p. 6.

Apart from initial terms, same as A001037.

Cf. A000225, A000740, A008683, A008965, A011782, A060223, A185700, A228369, A269134 A281013, A296302, A296373.

a059966 n = sum (map (\x -> a008683 (n `div` x) * a000225 x)
                     [d | d <- [1..n], mod n d == 0]) `div` n
-- Reinhard Zumkeller, Nov 18 2011

Table[1/n Apply[Plus, Map[(MoebiusMu[n/# ](2^# - 1)) &, Divisors[n]]], {n, 20}]
(* Second program: *)
Table[(1/n) DivisorSum[n, MoebiusMu[n/#] (2^# - 1) &], {n, 35}] (* Michael De Vlieger, Jul 22 2019 *)

from sympy import mobius, divisors
def A059966(n): return sum(mobius(n//d)*(2**d-1) for d in divisors(n,generator=True))//n # Chai Wah Wu, Feb 03 2022

G.f.: Product_{n>0} (1-q^n)^a(n) = 1-q-q^2-q^3-q^4-... = 2-1/(1-q).
Inverse Euler transform of A011782. - Alois P. Heinz, Jun 23 2018
G.f.: Sum_{k>=1} mu(k)*log((1 - x^k)/(1 - 2*x^k))/k. - Ilya Gutkovskiy, May 19 2019
a(n) ~ 2^n / n. - Vaclav Kotesovec, Aug 10 2019
Dirichlet g.f.: f(s+1)/zeta(s+1) - 1, where f(s) = Sum_{n>=1} 2^n/n^s. - Jianing Song, Nov 13 2021

Explicit formula from Paul D. Hanna, Apr 15 2002

Description corrected by Axel Kleinschmidt, Sep 15 2002

A102659 List of Lyndon words on {1,2} sorted first by length and then lexicographically.

Original entry on oeis.org

1, 2, 12, 112, 122, 1112, 1122, 1222, 11112, 11122, 11212, 11222, 12122, 12222, 111112, 111122, 111212, 111222, 112122, 112212, 112222, 121222, 122222, 1111112, 1111122, 1111212, 1111222, 1112112, 1112122, 1112212, 1112222, 1121122

Offset: 1

N. J. A. Sloane, Feb 03 2005

A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Bassino, J. Clement and C. Nicaud, The standard factorization of Lyndon words: an average point of view, Discrete Math. 290 (2005), 1-25.
Émilie Charlier, Manon Philibert, Manon Stipulanti, Nyldon words, arXiv:1804.09735 [math.CO], 2018. See Table 1.
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, 2012. - From _N. J. A. Sloane_, Dec 31 2012
Wikipedia, Lyndon word
Reinhard Zumkeller, Haskell programs for some sequences concerning Lyndon words
Index entries for sequences related to Lyndon words

Cf. A001037, A074650, A102660, A210584, A210585.
The "co" version is A329318.
A triangular version is A296657.
A sequence listing all Lyndon compositions is A294859.
Numbers whose binary expansion is Lyndon are A328596.
Length of the Lyndon factorization of the binary expansion is A211100.
Cf. A059966, A060223, A275692, A281013, A296373, A329131, A329313.

Haskell
```
cf. link.
```

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
Join@@Table[FromDigits/@Select[Tuples[{1,2},n],lynQ],{n,5}] (* Gus Wiseman, Nov 14 2019 *)

is_A102659(n)={ vecsort(d=digits(n))!=d&&for(i=1,#d-1, n>[1,10^(#d-i)]*divrem(n,10^i)&&return); fordiv(#d,L,L<#d && d==concat(Col(vector(#d/L,i,1)~*vecextract(d,2^L-1))~)&&return); !setminus(Set(d),[1,2])} \\ The last check is the least expensive one, but not useful if we test only numbers with digits {1,2}.
for(n=1,6,p=vector(n,i,10^(n-i))~;forvec(d=vector(n,i,[1,2]),is_A102659(m=d*p)&&print1(m","))) \\ One could use is_A102660 instead of is_A102659 here. - M. F. Hasler, Mar 08 2014

A102659 = A102660 intersect A007931 = A213969 intersect A239016. - M. F. Hasler, Mar 10 2014

More terms from Franklin T. Adams-Watters, Dec 14 2006

Definition improved by Reinhard Zumkeller, Mar 23 2012

A228369 Triangle read by rows in which row n lists the compositions (ordered partitions) of n in lexicographic order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 2, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 3, 1, 1, 3, 2, 4, 1, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Aug 28 2013

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is lexicographic. - Joerg Arndt, Sep 02 2013
The equivalent sequence for partitions is A026791.
Row n has length A001792(n-1).
Row sums give A001787, n >= 1.
The m-th composition has length A008687(m+1), m >= 1. - Andrey Zabolotskiy, Jul 19 2017

			Illustration of initial terms:
-----------------------------------
n  j       Diagram   Composition j
-----------------------------------
.               _
1  1           |_|   1;
.             _ _
2  1         | |_|   1, 1,
2  2         |_ _|   2;
.           _ _ _
3  1       | | |_|   1, 1, 1,
3  2       | |_ _|   1, 2,
3  3       |   |_|   2, 1,
3  4       |_ _ _|   3;
.         _ _ _ _
4  1     | | | |_|   1, 1, 1, 1,
4  2     | | |_ _|   1, 1, 2,
4  3     | |   |_|   1, 2, 1,
4  4     | |_ _ _|   1, 3,
4  5     |   | |_|   2, 1, 1,
4  6     |   |_ _|   2, 2,
4  7     |     |_|   3, 1,
4  8     |_ _ _ _|   4;
.
Triangle begins:
[1];
[1,1],[2];
[1,1,1],[1,2],[2,1],[3];
[1,1,1,1],[1,1,2],[1,2,1],[1,3],[2,1,1],[2,2],[3,1],[4];
[1,1,1,1,1],[1,1,1,2],[1,1,2,1],[1,1,3],[1,2,1,1],[1,2,2],[1,3,1],[1,4],[2,1,1,1],[2,1,2],[2,2,1],[2,3],[3,1,1],[3,2],[4,1],[5];
...

Joerg Arndt, Table of n, a(n) for n = 1..10000

Cf. A001511, A026791, A066099, A101211, A124734, A228351, A228525, A281013.

a228369 n = a228369_list !! (n - 1)
a228369_list = concatMap a228369_row [1..]
a228369_row 0 = []
a228369_row n
  | 2^k == 2 * n + 2 = [k - 1]
  | otherwise        = a228369_row (n `div` 2^k) ++ [k] where
    k = a007814 (n + 1) + 1
-- Peter Kagey, Jun 27 2016

Table[Sort[Join@@Permutations/@IntegerPartitions[n],OrderedQ[PadRight[{#1,#2}]]&],{n,5}] (* Gus Wiseman, Dec 14 2017 *)

gen_comp(n)=
{  /* Generate compositions of n as lists of parts (order is lex): */
    my(ct = 0);
    my(m, z, pt);
    \\ init:
    my( a = vector(n, j, 1) );
    m = n;
    while ( 1,
        ct += 1;
        pt = vector(m, j, a[j]);
        /* for A228369  print composition: */
        for (j=1, m, print1(pt[j],", ") );
\\        /* for A228525 print reversed (order is colex): */
\\        forstep (j=m, 1, -1, print1(pt[j],", ") );
        if ( m<=1,  return(ct) );  \\ current is last
        a[m-1] += 1;
        z = a[m] - 2;
        a[m] = 1;
        m += z;
    );
    return(ct);
}
for(n=1, 12, gen_comp(n) );
\\ Joerg Arndt, Sep 02 2013

a = [[[]], [[1]]]
for s in range(2, 9):
    a.append([])
    for k in range(1, s+1):
        for ss in a[s-k]:
            a[-1].append([k]+ss)
print(a)
# Andrey Zabolotskiy, Jul 19 2017

A296774 Triangle read by rows in which row n lists the compositions of n ordered first by length and then reverse-lexicographically.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 5, 4, 1, 3, 2, 2, 3, 1, 4, 3, 1, 1, 2, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 3, 3

Offset: 1

Gus Wiseman, Dec 20 2017

			Triangle of compositions begins:
(1),
(2),(11),
(3),(21),(12),(111),
(4),(31),(22),(13),(211),(121),(112),(1111),
(5),(41),(32),(23),(14),(311),(221),(212),(131),(122),(113),(2111),(1211),(1121),(1112),(11111).

Cf. A066099, A101211, A108730, A124734, A228369, A281013, A294859, A296302, A296656, A296772, A296773.

Table[Sort[Join@@Permutations/@IntegerPartitions[n],Or[Length[#1]

A296302 Number of aperiodic compositions of n with relatively prime parts. Number of compositions of n with relatively prime parts and relatively prime run-lengths.

Original entry on oeis.org

1, 0, 2, 5, 14, 24, 62, 114, 249, 480, 1022, 1978, 4094, 8064, 16348, 32520, 65534, 130512, 262142, 523270, 1048444, 2095104, 4194302, 8384316, 16777185, 33546240, 67108356, 134201398, 268435454, 536837136, 1073741822, 2147418240, 4294965244, 8589803520

Offset: 1

Gus Wiseman, Dec 11 2017

nonn

			The a(6) = 24 aperiodic compositions with relatively prime parts are:
(15), (51),
(114), (123), (132), (141), (213), (231), (312), (321), (411),
(1113), (1122), (1131), (1221), (1311), (2112), (2211), (3111),
(11112), (11121), (11211), (12111), (21111).

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A000005, A000740, A000837, A007427, A008683, A008965, A059966, A060223, A100953, A228369, A281013.

Table[DivisorSum[n,Function[d,MoebiusMu[n/d]*DivisorSum[d,MoebiusMu[#]*2^(d/#-1)&]]],{n,20}]

a = mu * mu * c, where * is Dirichlet convolution and c(n) = 2^(n-1).

A296372 Triangle read by rows: T(n,k) is the number of normal sequences of length n whose standard factorization into Lyndon words (aperiodic necklaces) has k factors.

Original entry on oeis.org

1, 1, 2, 4, 5, 4, 18, 31, 18, 8, 108, 208, 153, 56, 16, 778, 1700, 1397, 616, 160, 32, 6756, 15980, 14668, 7197, 2196, 432, 64, 68220, 172326, 171976, 93293, 31564, 7208, 1120, 128

Offset: 1

Gus Wiseman, Dec 11 2017

A finite sequence is normal if its union is an initial interval of positive integers.

			The T(3,2) = 5 normal sequences are {2,1,2}, {1,2,1}, {2,1,3}, {2,3,1}, {3,1,2}.
Triangle begins:
     1;
     1,     2;
     4,     5,     4;
    18,    31,    18,     8;
   108,   208,   153,    56,    16;
   778,  1700,  1397,   616,   160,    32;
  6756, 15980, 14668,  7197,  2196,   432,    64;

Andrew Howroyd, Rows n = 1..50 of triangle, flattened
Wikipedia, Lyndon word: Standard factorization

Row sums are A000670.
First column is A060223.
Cf. A000740, A001045, A008965, A019536, A059966, A074650, A185700, A228369, A232472, A277427, A281013, A296373.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
aperQ[q_]:=UnsameQ@@Table[RotateRight[q,k],{k,Length[q]}];
qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],neckQ[Take[q,#]]&&aperQ[Take[q,#]]&]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[qit[#]]===k&]],{n,5},{k,n}]

\\ here U(n,k) is A074650(n,k).
EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
U(n,k)={sumdiv(n, d, moebius(n/d) * k^d)/n}
A(n)={[Vecrev(p/y) | p<-sum(k=1, n, EulerMT(vector(n, n, y*U(n,k)))*sum(j=k, n, (-1)^(k-j)*binomial(j,k)))]}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 08 2018

Example and program corrected by Gus Wiseman, Dec 08 2018

A296373 Triangle T(n,k) = number of compositions of n whose factorization into Lyndon words (aperiodic necklaces) is of length k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 5, 3, 1, 1, 9, 12, 6, 3, 1, 1, 18, 21, 14, 6, 3, 1, 1, 30, 45, 27, 15, 6, 3, 1, 1, 56, 84, 61, 29, 15, 6, 3, 1, 1, 99, 170, 120, 67, 30, 15, 6, 3, 1, 1, 186, 323, 254, 136, 69, 30, 15, 6, 3, 1, 1, 335, 640, 510, 295, 142, 70, 30, 15, 6, 3, 1, 1

Offset: 1

Gus Wiseman, Dec 11 2017

			Triangle begins:
    1;
    1,   1;
    2,   1,   1;
    3,   3,   1,   1;
    6,   5,   3,   1,   1;
    9,  12,   6,   3,   1,   1;
   18,  21,  14,   6,   3,   1,   1;
   30,  45,  27,  15,   6,   3,   1,   1;
   56,  84,  61,  29,  15,   6,   3,   1,   1;
   99, 170, 120,  67,  30,  15,   6,   3,   1,   1;
  186, 323, 254, 136,  69,  30,  15,   6,   3,   1,   1;
  335, 640, 510, 295, 142,  70,  30,  15,   6,   3,   1,   1;

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Wikipedia, Lyndon word: Standard factorization

Cf. A000740, A001045, A008965, A019536, A059966, A060223, A185700, A228369, A232472, A277427, A281013, A296302, A296372.

neckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
aperQ[q_]:=UnsameQ@@Table[RotateRight[q,k],{k,Length[q]}];
qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],neckQ[Take[q,#]]&&aperQ[Take[q,#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[qit[#]]===k&]],{n,12},{k,n}]

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
A(n)=[Vecrev(p/y) | p<-EulerMT(y*vector(n, n, sumdiv(n, d, moebius(n/d) * (2^d-1))/n))]
{ my(T=A(12)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 01 2018

First column is A059966.

A296658 Length of the standard Lyndon word factorization of the first n terms of A000002.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 4, 3, 4, 5, 4, 3, 4, 3, 4, 5, 4, 5, 3, 3, 4, 5, 4, 5, 6, 5, 6, 4, 4, 5, 4, 4, 5, 6, 5, 6, 4, 4, 5, 4, 5, 6, 5, 6, 7, 6, 4, 5, 4, 4, 5, 6, 5, 6, 4, 4, 5, 4, 4, 5, 6, 5, 6, 7, 6, 7, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 5, 6, 7, 6

Offset: 1

Gus Wiseman, Dec 18 2017

nonn

			The standard Lyndon word factorization of (12211212212211211) is (122)(112122122)(112)(1)(1), so a(17) = 5.
The standard factorizations of initial terms of A000002:
1
12
122
122,1
122,1,1
122,112
122,112,1
122,11212
122,112122
122,112122,1
122,11212212
122,112122122
122,112122122,1
122,112122122,1,1
122,112122122,112
122,112122122,112,1
122,112122122,112,1,1
122,112122122,112,112
122,112122122,1121122
122,112122122,1121122,1

Frédérique Bassino, Julien Clement, and Cyril Nicaud, The standard factorization of Lyndon words: an average point of view, Discrete Mathematics, 290-1 (2005), 1-25.

Row lengths of A329315.
The "co-" version is A329362.
Cf. A000002, A001037, A027375, A060223, A088568, A102659, A211100, A281013, A288605, A296372, A296657, A296659, A328596, A329312, A329316, A329317.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],LyndonQ[Take[q,#]]&]];
kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],Part[q,-2],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]];
Table[Length[qit[Nest[kolagrow,1,n]]],{n,150}]

A294859 Triangle whose n-th row is the concatenated sequence of all Lyndon compositions of n in lexicographic order.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 1, 2, 1, 3, 4, 1, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 4, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 4, 1, 2, 3, 1, 3, 2, 1, 5, 2, 4, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 3, 2, 1

Offset: 1

Gus Wiseman, Dec 18 2017

			Triangle of Lyndon compositions begins:
(1),
(2),
(12),(3),
(112),(13),(4),
(1112),(113),(122),(14),(23),(5),
(11112),(1113),(1122),(114),(123),(132),(15),(24),(6),
(111112),(11113),(11122),(1114),(11212),(1123),(1132),(115),(1213),(1222),(124),(133),(142),(16),(223),(25),(34),(7).

Cf. A000740, A001037, A001045, A008965, A059966, A060223, A066099, A101211, A102659, A124734, A185700, A228369, A281013, A296302, A296373, A296656.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n],LyndonQ],OrderedQ[PadRight[{#1,#2}]]&],{n,7}]

Row n is a concatenation of A059966(n) Lyndon words with total length A000740(n).

A296772 Triangle read by rows in which row n lists the compositions of n ordered first by decreasing length and then reverse-lexicographically.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 2, 1, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 3, 4, 1, 3, 2, 2, 3, 1, 4, 5, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Dec 20 2017

The ordering of compositions in each row is consistent with the reverse-Mathematica ordering of expressions (cf. A124734).

Length of k-th composition is A124748(k-1)+1. - Andrey Zabolotskiy, Dec 20 2017

			Triangle of compositions begins:
(1),
(11),(2),
(111),(21),(12),(3),
(1111),(211),(121),(112),(31),(22),(13),(4),
(11111),(2111),(1211),(1121),(1112),(311),(221),(212),(131),(122),(113),(41),(32),(23),(14),(5).

Cf. A066099, A101211, A108730, A124734, A124748, A228369, A281013, A294859, A296302, A296656, A296773, A296774.

Table[Reverse[Sort[Join@@Permutations/@IntegerPartitions[n]]],{n,6}]

cp's OEIS Frontend

A059966 a(n) = (1/n) * Sum_{ d divides n } mu(n/d) * (2^d - 1).

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A102659 List of Lyndon words on {1,2} sorted first by length and then lexicographically.

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A228369 Triangle read by rows in which row n lists the compositions (ordered partitions) of n in lexicographic order.

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A296774 Triangle read by rows in which row n lists the compositions of n ordered first by length and then reverse-lexicographically.

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A296302 Number of aperiodic compositions of n with relatively prime parts. Number of compositions of n with relatively prime parts and relatively prime run-lengths.

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A296372 Triangle read by rows: T(n,k) is the number of normal sequences of length n whose standard factorization into Lyndon words (aperiodic necklaces) has k factors.

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A296373 Triangle T(n,k) = number of compositions of n whose factorization into Lyndon words (aperiodic necklaces) is of length k.

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