A000031 -id:A000031 - cp's OEIS frontend

A329138 Numbers whose prime signature is a necklace.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A304678 in having 1350 = 2^1 * 3^3 * 5^2. First differs from A316529 in having 150 = 2^1 * 3^1 * 5^2.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their prime signatures begins:
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   6: (1,1)
   7: (1)
   8: (3)
   9: (2)
  10: (1,1)
  11: (1)
  13: (1)
  14: (1,1)
  15: (1,1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  21: (1,1)
  22: (1,1)

Complement of A329142.
Binary necklaces are A000031.
Necklace compositions are A008965.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is aperiodic are A329139.
Cf. A001037, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329140.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[2,100],neckQ[Last/@FactorInteger[#]]&]

A051841 Number of binary Lyndon words with an even number of 1's.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 14, 28, 48, 93, 165, 315, 576, 1091, 2032, 3855, 7252, 13797, 26163, 49929, 95232, 182361, 349350, 671088, 1290240, 2485504, 4792905, 9256395, 17894588, 34636833, 67106816, 130150493, 252641280, 490853403, 954429840, 1857283155, 3616800768, 7048151355, 13743869130, 26817356775

Offset: 1

Frank Ruskey, Dec 13 1999

Also number of trace 0 irreducible polynomials over GF(2).

Also number of trace 0 Lyndon words over GF(2).

			a(5) = 3 = |{ 00011, 00101, 01111 }|.

May, Robert M. "Simple mathematical models with very complicated dynamics." Nature, Vol. 261, June 10, 1976, pp. 459-467; reprinted in The Theory of Chaotic Attractors, pp. 85-93. Springer, New York, NY, 2004. The sequences listed in Table 2 are A000079, A027375, A000031, A001037, A000048, A051841. - N. J. A. Sloane, Mar 17 2019

T. D. Noe, Table of n, a(n) for n = 1..300
F. Ruskey, Number of q-ary Lyndon words with given trace mod q
F. Ruskey, Number of Lyndon words over GF(q) with given trace
Index entries for sequences related to Lyndon words

Same as A001037 - A000048. Same as A042980 + A042979.
Cf. A027750, A008683.
Cf. A000010.

a051841 n = (sum $ zipWith (\u v -> gcd 2 u * a008683 u * 2 ^ v)
             ds $ reverse ds) `div` (2 * n) where ds = a027750_row n
-- Reinhard Zumkeller, Mar 17 2013

a[n_] := Sum[GCD[d, 2]*MoebiusMu[d]*2^(n/d), {d, Divisors[n]}]/(2n);
Table[a[n], {n, 1, 32}]
(* Jean-François Alcover, May 14 2012, from formula *)

L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%2==0, L(n, k), 0 ) ) / n;
vector(33,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

a(n) = 1/(2*n)*Sum_{d|n} gcd(d,2)*mu(d)*2^(n/d).
a(n) ~ 2^(n-1) / n. - Vaclav Kotesovec, May 31 2019
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = 1/(2*n)*Sum_{k=1..n} gcd(gcd(n,k),2)*mu(gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} gcd(n/gcd(n,k),2)*mu(n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)

A323858 Number of toroidal necklaces of positive integers summing to n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 31, 44, 90, 154, 296, 524, 1035, 1881, 3636, 6869, 13208, 25150, 48585, 93188, 180192, 347617, 673201, 1303259, 2529740, 4910708, 9549665, 18579828, 36192118, 70540863, 137620889, 268655549, 524873503, 1026068477, 2007178821, 3928564237

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (necklace) case is A008965.

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.

			Inequivalent representatives of the a(6) = 31 toroidal necklaces:
  6  15  24  33  114  123  132  222  1113  1122  1212  11112  111111
.
  1  2  3  11  11  12  12  111
  5  4  3  13  22  12  21  111
.
  1  1  1  2  11
  1  2  3  2  11
  4  3  2  2  11
.
  1  1  1
  1  1  2
  1  2  1
  3  2  2
.
  1
  1
  1
  1
  2
.
  1
  1
  1
  1
  1
  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A000670, A008965, A059966, A101509, A179043, A184271.

Cf. A323859, A323861, A323865, A323866, A323870.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Join@@Table[Select[ptnmats[k],neckmatQ],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]],{n,10}]

U(n,m,k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * subst(k, x, x^lcm(c,d))^(n*m/lcm(c, d))));
a(n)={if(n < 1, n==0, sum(i=1, n, sum(j=1, n\i, polcoef(U(i, j, x/(1-x) + O(x*x^n)), n))))} \\ Andrew Howroyd, Aug 18 2019

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

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

Original entry on oeis.org

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).

A053656 Number of cyclic graphs with oriented edges on n nodes (up to symmetry of dihedral group).

Original entry on oeis.org

1, 2, 2, 4, 4, 9, 10, 22, 30, 62, 94, 192, 316, 623, 1096, 2122, 3856, 7429, 13798, 26500, 49940, 95885, 182362, 350650, 671092, 1292762, 2485534, 4797886, 9256396, 17904476, 34636834, 67126282, 130150588, 252679832, 490853416

Offset: 1

Jeb F. Willenbring (jwillenb(AT)ucsd.edu), Feb 14 2000

Also number of bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over.
a(n) is also the number of frieze patterns generated by filling a 1 X n block with n copies of an asymmetric motif (where the copies are chosen from original motif or a 180-degree rotated copy) and then repeating the block by translation to produce an infinite frieze pattern. (Pisanski et al.)
a(n) is also the number of minimal fibrations of a bidirectional n-cycle over the 2-bouquet up to precompositions with automorphisms of the n-cycle. (Boldi et al.) - Sebastiano Vigna, Jan 08 2018

			2 at n=3 because there are two such cycles. On (o -> o -> o ->) and (o -> o <- o ->).

Jeb F. Willenbring, A stability result for a Hilbert series of O_n(C) invariants.

Seiichi Manyama, Table of n, a(n) for n = 1..3334
Rémi Cocou Avohou, Joseph Ben Geloun, and Reiko Toriumi, Counting U(N)^{⊗r} ⊗ O(N)^{⊗q} invariants and tensor model observables, Eur. Phys. J. C 84, 839 (2024), see pp. 11, 27; Preprint arXiv:2404.16404 [hep-th], 2024. See pp. 18, 49.
Paolo Boldi and Sebastiano Vigna, Fibrations of Graphs, Discrete Math., 243 (2002), 21-66.
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366. See Table 8.
T. Pisanski, D. Schattschneider, and B. Servatius, Applying Burnside's lemma to a one-dimensional Escher problem, Math. Mag., 79 (2006), 167-180.
Jeb F. Willenbring, Home page.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264. See Tables 1 and 2 (and text).
Index entries for sequences related to bracelets.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

v:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*2^(n/k); od: t1; end;
h:=n-> if n mod 2 = 0 then (n/2)*2^(n/2); else 0; fi;
A053656:=n->(v(n)+h(n))/(2*n); # N. J. A. Sloane, Nov 11 2006

a[n_] := Total[ EulerPhi[#]*2^(n/#)& /@ Divisors[n]]/(2n) + 2^(n/2-2)(1-Mod[n, 2]); Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Nov 21 2011 *)

a(n)={(sumdiv(n, d, eulerphi(d)*2^(n/d))/n + if(n%2==0, 2^(n/2-1)))/2} \\ Andrew Howroyd, Jun 16 2021

G.f.: x/(1-x) + x^2/(2*(1-2*x^2)) + Sum_{n >= 1} (x^(2*n)/(2*n)) * Sum_{ d divides n } phi(d)/(1-x^d)^(2*n/d), or x^2/(2*(1-2*x^2)) - Sum_{n >= 1} phi(n)*log(1-2*x^n)/(2*n). [corrected and extended by Andrey Zabolotskiy, Oct 17 2017]

a(n) = A000031(n)/2 + (if n even) 2^(n/2-2).

More terms and additional comments from Christian G. Bower, Dec 13 2001

A323861 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.

Original entry on oeis.org

2, 1, 1, 2, 2, 2, 3, 9, 9, 3, 6, 27, 54, 27, 6, 9, 99, 335, 335, 99, 9, 18, 326, 2182, 4050, 2182, 326, 18, 30, 1161, 14523, 52377, 52377, 14523, 1161, 30, 56, 4050, 99858, 698535, 1342170, 698535, 99858, 4050, 56, 99, 14532, 698870, 9586395, 35790267, 35790267, 9586395, 698870, 14532, 99

Offset: 1

Gus Wiseman, Feb 04 2019

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.

			Table begins:
        1    2    3    4
    ------------------------
  1: |  2    1    2    3
  2: |  1    2    9   27
  3: |  2    9   54  335
  4: |  3   27  335 4050
Inequivalent representatives of the A(3,2) = 9 aperiodic toroidal necklaces:
  [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1]
  [0 0 1] [0 1 1] [0 1 0] [0 1 1] [1 0 1] [1 1 0] [1 1 1] [1 0 1] [1 1 1]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.
Andrew Howroyd, GAP Program Code

First and last columns are A001037. Main diagonal is A323872.
Cf. A000031, A000740, A027375, A059966, A179043, A184271.
Cf. A323859, A323860, A323865, A323866, A323871.

# See link for code.
for n in [1..8] do for k in [1..8] do Print(A323861(n,k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019

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-k]&/@Tuples[{0,1},(n-k)*k],And[apermatQ[#],neckmatQ[#]]&]],{n,6},{k,n-1}]

Terms a(37) and beyond from Andrew Howroyd, Aug 21 2019

A323865 Number of aperiodic binary toroidal necklaces of size n.

Original entry on oeis.org

1, 2, 2, 4, 8, 12, 36, 36, 114, 166, 396, 372, 1992, 1260, 4644, 8728, 20310, 15420, 87174, 55188, 314064, 399432, 762228, 729444, 5589620, 4026522, 10323180, 19883920, 57516048, 37025580, 286322136, 138547332, 805277760, 1041203944, 2021145660, 3926827224

Offset: 0

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. 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(6) = 36 aperiodic necklaces:
  000001  000011  000101  000111  001011  001101  001111  010111  011111
.
  000  000  001  001  001  001  001  011  011
  001  011  010  011  101  110  111  101  111
.
  00  00  00  00  00  01  01  01  01
  00  01  01  01  11  01  01  10  11
  01  01  10  11  01  10  11  11  11
.
  0  0  0  0  0  0  0  0  0
  0  0  0  0  0  0  0  1  1
  0  0  0  0  1  1  1  0  1
  0  0  1  1  0  1  1  1  1
  0  1  0  1  1  0  1  1  1
  1  1  1  1  1  1  1  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A001037, A027375, A179043, A184271, A323351.

Cf. A323858, A323859, A323860, A323861, A323864, 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]]}]];
zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
Table[If[n==0,1,Length[Union[First/@matcyc/@Select[zaz[n],And[apermatQ[#],neckmatQ[#]]&]]]],{n,0,10}]

a(n) = Sum_{d|n} A323861(d, n/d) for n > 0. - Andrew Howroyd, Aug 21 2019

Terms a(19) and beyond from Andrew Howroyd, Aug 21 2019

A329315 Irregular triangle read by rows where row n gives the sequence of lengths of components of the Lyndon factorization of the first n terms of A000002.

Original entry on oeis.org

1, 2, 3, 3, 1, 3, 1, 1, 3, 3, 3, 3, 1, 3, 5, 3, 6, 3, 6, 1, 3, 8, 3, 9, 3, 9, 1, 3, 9, 1, 1, 3, 9, 3, 3, 9, 3, 1, 3, 9, 3, 1, 1, 3, 9, 3, 3, 3, 9, 7, 3, 9, 7, 1, 3, 9, 9, 3, 9, 9, 1, 3, 9, 9, 1, 1, 3, 9, 9, 3, 3, 9, 9, 3, 1, 3, 9, 14, 3, 9, 15, 3, 9, 15, 1, 3

Offset: 1

Gus Wiseman, Nov 11 2019

There are no repeated rows, as row n has sum n.
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 word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
It appears that some numbers (such as 4) never appear in the sequence.

			Triangle begins:
   1: (1)
   2: (2)
   3: (3)
   4: (3,1)
   5: (3,1,1)
   6: (3,3)
   7: (3,3,1)
   8: (3,5)
   9: (3,6)
  10: (3,6,1)
  11: (3,8)
  12: (3,9)
  13: (3,9,1)
  14: (3,9,1,1)
  15: (3,9,3)
  16: (3,9,3,1)
  17: (3,9,3,1,1)
  18: (3,9,3,3)
  19: (3,9,7)
  20: (3,9,7,1)
For example, the first 10 terms of A000002 are (1221121221), with Lyndon factorization (122)(112122)(1), so row 10 is (3,6,1).

Row lengths are A296658.
The reversed version is A329316.
Cf. A000002, A000031, A001037, A027375, A059966, A060223, A088568, A102659, A211100, A288605, A296372, A329314, A329317, A329325.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#1]]&]]]];
kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],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]]];
kol[n_Integer]:=Nest[kolagrow,{1},n-1];
Table[Length/@lynfac[kol[n]],{n,100}]

A210109 Number of 3-divided binary sequences (or words) of length n.

Original entry on oeis.org

0, 0, 0, 2, 7, 23, 54, 132, 290, 634, 1342, 2834, 5868, 12140, 24899, 50929, 103735, 210901, 427623, 865910, 1750505, 3535098, 7131321, 14374647, 28952661, 58280123, 117248217, 235770302, 473897980, 952183214, 1912535827, 3840345963, 7709282937, 15472242645, 31045402788, 62280978042

Offset: 1

N. J. A. Sloane, Mar 17 2012

nonn

A binary sequence (or word) W of length n is 3-divided if it can be written as a concatenation W = XYZ such that XYZ is strictly earlier in lexicographic order than any of the five permutations  XZY, ZYX, YXZ, YZX, ZXY.
More generally, fix an alphabet A = {0,1,2,...,a-1}.
Define lexicographic order on words over A in the obvious way: for single letters, i < j is the natural order; for words U, V, we set U < V iff u_i < v_i at the first place where they differ; also U < UV if V is nonempty, etc.
Then a word U over A is "k-divided over A" if it can be written as U = X_1 X_2 ... X_k in such a way that X is strictly less in lexicographic order than X_pi_1 X_pi_2 ... X_pi_k for all nontrivial permutations pi of [1..k].
All 2^n binary words are 1-divided. For 2-divided words see A209970.
"k-divisible" would sound better to me than "k-divided", but I have followed Lothaire and Pirillo-Varricchio in using the latter term. Neither reference gives this sequence.

			The two 3-divisible binary words of length 4 and the seven of length 5 are as follows. The periods indicate the division w = x.y.z. For example, 0.01.1 is 3-divided since 0011 < all of 0101, 1010, 0101, 1001, 0110.
0.01.1
0.10.1
0.001.1
0.010.1
0.01.10
0.01.11
0.100.1
0.10.11
0.110.1

M. Lothaire, Combinatorics on words. A collective work by Dominique Perrin, Jean Berstel, Christian Choffrut, Robert Cori, Dominique Foata, Jean Eric Pin, Guiseppe Pirillo, Christophe Reutenauer, Marcel-P. Schützenberger, Jacques Sakarovitch and Imre Simon. With a foreword by Roger Lyndon. Edited and with a preface by Perrin. Encyclopedia of Mathematics and its Applications, 17. Addison-Wesley Publishing Co., Reading, Mass., 1983. xix+238 pp. ISBN: 0-201-13516-7, MR0675953 (84g:05002). See p. 144.

Michael S. Branicky, Python program using bitwise operations
Giuseppe Pirillo and Stefano Varricchio, Some combinatorial properties of infinite words and applications to semigroup theory. Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993). Discrete Math. 153 (1996), no. 1-3, pages 239-251, MR1394958 (98f:05018).

Number of k-divided words of length n over alphabet of size A:
A=2, k=2,3,4,5: A209970 (and A209919, A000031, A001037), A210109 (and A210145), A210321, A210322;
A=3, k=2,3,4,5: A210323 (and A001867, A027376), A210324, A210325, A210326;
A=4, k=2,3,4: A210424 (and A001868, A027377), A210425, A210426.

# see link for faster, bit-based version
from itertools import product
def is3div(b):
    for i in range(1, len(b)-1):
        for j in range(i+1, len(b)):
            X, Y, Z = b[:i], b[i:j], b[j:]
            if all(b < bp for bp in [Z+Y+X, Z+X+Y, Y+Z+X, Y+X+Z, X+Z+Y]):
                return True
    return False
def a(n): return sum(is3div("".join(b)) for b in product("01", repeat=n))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Aug 27 2021

Is there a formula or recurrence?

Values confirmed and a(30)-a(31) by David Applegate, Mar 19 2012

a(32)-a(36) from Michael S. Branicky, Aug 27 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

A329138 Numbers whose prime signature is a necklace.

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A051841 Number of binary Lyndon words with an even number of 1's.

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A323858 Number of toroidal necklaces of positive integers summing to n.

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

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A053656 Number of cyclic graphs with oriented edges on n nodes (up to symmetry of dihedral group).

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A323861 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.

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A323865 Number of aperiodic binary toroidal necklaces of size n.

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