A008965 -id:A008965 - cp's OEIS frontend

A333940 Number of Lyndon factorizations of the k-th composition in standard order.

1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 5, 1, 2, 2, 4, 1, 4, 2, 7, 1, 2, 1, 4, 1, 2, 1, 7, 1, 2, 2, 4, 2, 5, 2, 7, 1, 2, 3, 9, 2, 5, 2, 12, 1, 2, 1, 4, 1, 2, 2, 7, 1, 2, 1, 4, 1, 2, 1, 11, 1, 2, 2, 4, 2, 5, 2, 7, 1, 4, 4, 11, 2, 5, 2, 12, 1, 2, 2, 4, 1, 7

Offset: 0

Gus Wiseman, Apr 13 2020

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon factorization of a composition c is a multiset of compositions whose Lyndon product is c.
A composition of n is a finite sequence of positive integers summing to n. 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.
Also the number of multiset partitions of the Lyndon-word factorization of the n-th composition in standard order.

			We have  a(300) = 5, because the 300th composition (3,2,1,3) has the following Lyndon factorizations:
  ((3,2,1,3))
  ((1,3),(3,2))
  ((2),(3,1,3))
  ((3),(2,1,3))
  ((2),(3),(1,3))

The dual version is A333765.
Binary necklaces are counted by A000031.
Necklace compositions are counted by A008965.
Necklaces covering an initial interval are counted by A019536.
Lyndon compositions are counted by A059966.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a necklace are A329138.
Length of Lyndon factorization of binary expansion is A211100.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of co-Lyndon factorization of reversed binary expansion is A329326.
Length of Lyndon factorization of reversed binary expansion is A329313.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Length of Lyndon factorization is A329312.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Dealing are counted by A333939.
- Reversed necklaces are A333943.
- Length of co-Lyndon factorization is A334029.
- Combinatory separations are A334030.
Cf. A000740, A001037, A034691, A060223, A269134, A292884, A328596.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynprod[]:={};lynprod[{},b_List]:=b;lynprod[a_List,{}]:=a;lynprod[a_List]:=a;
lynprod[{x_,a___},{y_,b___}]:=Switch[Ordering[If[x=!=y,{x,y},{lynprod[{a},{x,b}],lynprod[{x,a},{b}]}]],{2,1},Prepend[lynprod[{a},{y,b}],x],{1,2},Prepend[lynprod[{x,a},{b}],y]];
lynprod[a_List,b_List,c__List]:=lynprod[a,lynprod[b,c]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
dealings[q_]:=Union[Function[ptn,Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];
Table[Length[Select[dealings[stc[n]],lynprod@@#==stc[n]&]],{n,0,100}]

For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A034691(n).

A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 11, 16, 24, 36, 52, 73, 101, 135, 184, 244, 321, 418, 543, 694, 889, 1127, 1427, 1789, 2242, 2787, 3463, 4276, 5271, 6465, 7921, 9655, 11756, 14254, 17262, 20830, 25102, 30152, 36172, 43270, 51691, 61594, 73300, 87023, 103189, 122099, 144296, 170193, 200497

Offset: 0

Gus Wiseman, Feb 12 2022

nonn

			The a(4) = 1 through a(9) = 16 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (2211)   (331)    (422)      (522)
                (21111)  (511)    (611)      (711)
                         (3211)   (3221)     (3321)
                         (22111)  (3311)     (4221)
                         (31111)  (4211)     (4311)
                                  (22211)    (5211)
                                  (32111)    (22221)
                                  (41111)    (32211)
                                  (221111)   (33111)
                                  (2111111)  (42111)
                                             (51111)
                                             (222111)
                                             (321111)
                                             (2211111)
                                             (3111111)
For example, the partition x = (2,1,1,1,1) has the permutation (1,1,2,1,1), with runs (1,1), (2), (1,1), which are not all distinct, so x is counted under a(6).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A144300.
The version for normal multisets is A283353.
The Heinz numbers of these partitions are A351201.
The complement is counted by A351204.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions, ranked by A333489.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A238130 or A238279, A325545, A329746, A350842, A351003, A351004, A351291.

Table[Length[Select[IntegerPartitions[n],MemberQ[Permutations[#],_?(!UnsameQ@@Split[#]&)]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
from itertools import permutations, groupby
from collections import Counter
def A351203(n):
    c = 0
    for s, p in partitions(n,size=True):
        for q in permutations(Counter(p).elements(),s):
            if max(Counter(tuple(g) for k, g in groupby(q)).values(),default=0) > 1:
                c += 1
                break
    return c # Chai Wah Wu, Oct 16 2023

a(n) = A000041(n) - A351204(n). - Andrew Howroyd, Jan 27 2024

a(26) onwards from Andrew Howroyd, Jan 27 2024

A318726 Number of integer compositions of n that have only one part or whose consecutive parts are indivisible and the last and first part are also indivisible.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 3, 8, 13, 12, 23, 27, 56, 64, 100, 150, 216, 325, 459, 700, 1007, 1493, 2186, 3203, 4735, 6929, 10243, 14952, 22024, 32366, 47558, 69906, 102634, 150984, 221713, 325919, 478842, 703648, 1034104, 1519432, 2233062, 3281004, 4821791, 7085359

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(10) = 13 compositions:
  (10)
  (7,3) (3,7) (6,4) (4,6)
  (5,3,2) (5,2,3) (3,5,2) (3,2,5) (2,5,3) (2,3,5)
  (3,2,3,2) (2,3,2,3)
The a(11) = 12 compositions:
  (11)
  (9,2) (2,9) (8,3) (3,8) (7,4) (4,7) (6,5) (5,6)
  (5,2,4) (4,5,2) (2,4,5)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000740, A008965, A167606, A285573, A296302, A303362, A304713, A316476, A318727.

Table[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,({_,x_,y_,_}/;Divisible[x,y])|({y_,_,x_}/;Divisible[x,y])]&]//Length,{n,20}]

b(n,k,pred)={my(M=matrix(n,n)); for(n=1, n, M[n,n]=pred(k,n); for(j=1, n-1, M[n,j]=sum(i=1, n-j, if(pred(i,j), M[n-j,i], 0)))); sum(i=1, n, if(pred(i,k), M[n,i], 0))}
a(n)={1 + sum(k=1, n-1, b(n-k, k, (i,j)->i%j<>0))} \\ Andrew Howroyd, Sep 08 2018

a(n) = A328598(n) + 1. - Gus Wiseman, Nov 04 2019

a(21)-a(28) from Robert Price, Sep 08 2018
Terms a(29) and beyond from Andrew Howroyd, Sep 08 2018
Name corrected by Gus Wiseman, Nov 04 2019

A318728 Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose adjacent parts (including the last with first) are coprime.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 22, 34, 58, 95, 168, 280, 492, 853, 1508, 2648, 4715, 8350, 14924, 26643, 47794, 85779, 154475, 278323, 502716, 908913, 1646206, 2984547, 5418653, 9847190, 17916001, 32625618, 59470540, 108493150, 198094483, 361965239, 661891580, 1211162271

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(7) = 13 cyclic compositions with adjacent parts coprime:
  7,
  16, 25, 34,
  115,
  1114, 1213, 1132, 1123,
  11113, 11212,
  111112,
  1111111.

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

Cf. A000740, A000837, A008965, A059966, A100953, A167606, A296302, A318726, A318727, A328597.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&And@@CoprimeQ@@@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)->gcd(i,j)==1))); vector(n, n, (n > 1) + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

a(n) = A328597(n) + 1 for n > 1. - Andrew Howroyd, Oct 27 2019

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

Name corrected by Gus Wiseman, Nov 04 2019

A318729 Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose consecutive parts (including the last with first) are indivisible.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 4, 6, 6, 8, 11, 19, 21, 30, 41, 59, 79, 112, 157, 219, 305, 430, 605, 860, 1210, 1727, 2424, 3463, 4905, 7001, 9954, 14211, 20271, 28980, 41392, 59254, 84800, 121540, 174163, 249932, 358578, 515091, 739933, 1063827, 1529767, 2201383

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(13) = 11 cyclic compositions with successive parts indivisible:
  (13)
  (2,11) (3,10) (4,9) (5,8) (6,7)
  (2,4,7) (2,6,5) (2,8,3) (3,6,4)
  (2,3,5,3)

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

Cf. A000740, A008965, A059966, A167606, A285573, A303362, A304713, A316476, A318726, A318727, A318728, A318730, A328600.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&And@@Not/@Divisible@@@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))); vector(n, n, 1 + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

a(n) = A328600(n) + 1. - Andrew Howroyd, Oct 27 2019

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

Name corrected by Gus Wiseman, Nov 04 2019

A323870 Number of toroidal necklaces of size n whose entries cover an initial interval of positive integers.

Original entry on oeis.org

1, 4, 10, 61, 218, 3136, 13514, 272998, 2362439, 40899248, 295024106, 14045787790, 81055130522, 3040383719360, 61408850927732, 1661142088494553, 15337737297545402, 1128511554421317128, 9768588138876674858, 803306338873366385030, 15452347618762680757428

Offset: 1

Gus Wiseman, Feb 04 2019

nonn

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. Alternatively, a toroidal necklace is a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns.

			The a(3) = 10 toroidal necklaces:
  [1 2 3] [1 3 2] [1 2 2] [1 1 2] [1 1 1]
.
  [1] [1] [1] [1] [1]
  [2] [3] [2] [1] [1]
  [3] [2] [2] [2] [1]

Andrew Howroyd, Table of n, a(n) for n = 1..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000670, A008965, A060223.

Cf. A323858, A323859, A323866, A323868, A323871.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[nrmmats[n],neckmatQ]],{n,6}]

U(n,m,k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * k^(n*m/lcm(c, d))));
R(v)={sum(n=1, #v, sum(k=1, n, (-1)^(n-k)*binomial(n,k)*v[k]))}
a(n)={if(n < 1, n==0, R(vector(n, k, sumdiv(n, d, U(d, n/d, k))) ))} \\ Andrew Howroyd, Aug 18 2019

Terms a(9) and beyond from Andrew Howroyd, Aug 18 2019

A325546 Number of compositions of n with weakly increasing differences.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 19, 28, 41, 62, 87, 120, 170, 228, 303, 408, 534, 689, 899, 1145, 1449, 1842, 2306, 2863, 3571, 4398, 5386, 6610, 8039, 9716, 11775, 14157, 16938, 20293, 24166, 28643, 33995, 40134, 47199, 55540, 65088, 75994, 88776, 103328, 119886, 139126

Offset: 0

Gus Wiseman, May 10 2019

nonn

Also compositions of n whose plot is concave-up.
A composition of n is a finite sequence of positive integers summing to n.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (112)   (41)     (42)
                    (211)   (113)    (51)
                    (1111)  (212)    (114)
                            (311)    (123)
                            (1112)   (213)
                            (2111)   (222)
                            (11111)  (312)
                                     (321)
                                     (411)
                                     (1113)
                                     (2112)
                                     (3111)
                                     (11112)
                                     (21111)
                                     (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A007294, A008965, A070211 (concave-down compositions), A173258, A175342, A240026, A325360, A325545, A325547, A325548, A325552, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@Differences[#]&]],{n,0,15}]

\\ Row sums of R(n) give A007294 (=breakdown by width).
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-w-1)\t + 1, v[i-w-(k-1)*t]))); Mat(L)}
seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M,n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j,w])));  x^i/(1 - x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ Andrew Howroyd, Aug 28 2019

More terms from Alois P. Heinz, May 11 2019

A325680 Number of compositions of n such that every distinct circular subsequence has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 8, 14, 16, 29, 24, 42, 46, 78, 66, 146, 133, 242, 208, 386, 352, 620, 494, 948, 842, 1447

Offset: 0

Gus Wiseman, May 13 2019

A composition of n is a finite sequence of positive integers summing to n.

A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (1111)  (41)     (42)      (43)       (44)
                            (11111)  (51)      (52)       (53)
                                     (222)     (61)       (62)
                                     (111111)  (124)      (71)
                                               (142)      (125)
                                               (214)      (152)
                                               (241)      (215)
                                               (412)      (251)
                                               (421)      (512)
                                               (1111111)  (521)
                                                          (2222)
                                                          (11111111)

Cf. A000079, A008965, A108917, A143823, A169942, A276024.

Cf. A325545, A325676, A325682, A325685, A325687, A325688.

subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@subalt[#]&]],{n,0,15}]

a(18)-a(25) from Robert Price, Jun 19 2021

A329324 Number of Lyndon compositions of n whose reverse is not a co-Lyndon composition.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 7, 16, 37, 76, 166, 328, 669, 1326, 2626, 5138, 10104, 19680, 38442, 74822, 145715, 283424, 551721, 1073224

Offset: 1

Gus Wiseman, Nov 11 2019

A Lyndon composition of n is a finite sequence summing to n that is lexicographically strictly less than all of its cyclic rotations. A co-Lyndon composition of n is a finite sequence summing to n that is lexicographically strictly greater than all of its cyclic rotations.

			The a(6) = 1 through a(9) = 16 compositions:
  (132)  (142)   (143)    (153)
         (1132)  (152)    (162)
                 (1142)   (243)
                 (1232)   (1143)
                 (1322)   (1152)
                 (11132)  (1242)
                 (11312)  (1332)
                          (1422)
                          (11142)
                          (11232)
                          (11322)
                          (11412)
                          (12132)
                          (111132)
                          (111312)
                          (112212)

Lyndon and co-Lyndon compositions are counted by A059966.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Lyndon compositions that are not weakly increasing are A329141.
Cf. A000740, A001037, A008965, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Array[Union[{RotateRight[q,#1],q}]=={RotateRight[q,#1],q}&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],lynQ[#]&&!colynQ[Reverse[#]]&]],{n,15}]

a(21)-a(25) from Robert Price, Jun 20 2021

A333765 Number of co-Lyndon factorizations of the k-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 4, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 2, 4, 4, 7, 7, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 5, 2, 5, 2, 4, 4, 9, 4, 7, 7, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 4, 1

Offset: 0

Gus Wiseman, Apr 13 2020

nonn

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon factorization of a composition c is a multiset of compositions whose co-Lyndon product is c.
A composition of n is a finite sequence of positive integers summing to n. 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.
Also the number of multiset partitions of the co-Lyndon-word factorization of the n-th composition in standard order.

			The a(54) = 5, a(61) = 7, and a(237) = 9 factorizations:
  ((1,2,1,2))      ((1,1,1,2,1))        ((1,1,2,1,2,1))
  ((1),(2,1,2))    ((1),(1,1,2,1))      ((1),(1,2,1,2,1))
  ((1,2),(2,1))    ((1,1),(1,2,1))      ((1,1),(2,1,2,1))
  ((2),(1,2,1))    ((2,1),(1,1,1))      ((1,2,1),(1,2,1))
  ((1),(2),(2,1))  ((1),(1),(1,2,1))    ((2,1),(1,1,2,1))
                   ((1),(1,1),(2,1))    ((1),(1),(2,1,2,1))
                   ((1),(1),(1),(2,1))  ((1,1),(2,1),(2,1))
                                        ((1),(2,1),(1,2,1))
                                        ((1),(1),(2,1),(2,1))

The dual version is A333940.
Binary necklaces are counted by A000031.
Necklace compositions are counted by A008965.
Necklaces covering an initial interval are counted by A019536.
Lyndon compositions are counted by A059966.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a necklace are A329138.
Length of Lyndon factorization of binary expansion is A211100.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of co-Lyndon factorization of reversed binary expansion is A329326.
Length of Lyndon factorization of reversed binary expansion is A329313.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Length of Lyndon factorization is A329312.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Dealings are counted by A333939.
- Reversed necklaces are A333943.
- Length of co-Lyndon factorization is A334029.
- Combinatory separations are A334030.
Cf. A000740, A001037, A034691, A060223, A211100, A269134, A292884, A328596.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
colynprod[]:={};colynprod[{},b_List]:=b;colynprod[a_List,{}]:=a;colynprod[a_List]:=a;
colynprod[{x_,a___},{y_,b___}]:=Switch[Ordering[If[x=!=y,{x,y},{colynprod[{a},{x,b}],colynprod[{x,a},{b}]}]],{1,2},Prepend[colynprod[{a},{y,b}],x],{2,1},Prepend[colynprod[{x,a},{b}],y]];
colynprod[a_List,b_List,c__List]:=colynprod[a,colynprod[b,c]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
dealings[q_]:=Union[Function[ptn,Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];
Table[Length[Select[dealings[stc[n]],colynprod@@#==stc[n]&]],{n,0,100}]

For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A034691(n).

cp's OEIS Frontend

A333940 Number of Lyndon factorizations of the k-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A318726 Number of integer compositions of n that have only one part or whose consecutive parts are indivisible and the last and first part are also indivisible.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A318728 Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose adjacent parts (including the last with first) are coprime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A318729 Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose consecutive parts (including the last with first) are indivisible.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A323870 Number of toroidal necklaces of size n whose entries cover an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A325546 Number of compositions of n with weakly increasing differences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs