A065609 -id:A065609 - cp's OEIS frontend

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

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[#]]&]

A334274 Numbers k such that the k-th composition in standard order is both a necklace and a reversed co-necklace.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 48, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 127, 128, 136, 144, 160, 164, 168, 170, 192, 200, 204, 208, 212, 216

Offset: 1

Gus Wiseman, Apr 25 2020

nonn

Also numbers whose binary expansion is both a reversed necklace and a co-necklace.
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 co-necklace necklaces begins:
    0: ()            31: (1,1,1,1,1)      100: (1,3,3)
    1: (1)           32: (6)              104: (1,2,4)
    2: (2)           36: (3,3)            106: (1,2,2,2)
    3: (1,1)         40: (2,4)            108: (1,2,1,3)
    4: (3)           42: (2,2,2)          112: (1,1,5)
    6: (1,2)         48: (1,5)            116: (1,1,2,3)
    7: (1,1,1)       52: (1,2,3)          118: (1,1,2,1,2)
    8: (4)           54: (1,2,1,2)        120: (1,1,1,4)
   10: (2,2)         56: (1,1,4)          122: (1,1,1,2,2)
   12: (1,3)         58: (1,1,2,2)        124: (1,1,1,1,3)
   14: (1,1,2)       60: (1,1,1,3)        126: (1,1,1,1,1,2)
   15: (1,1,1,1)     62: (1,1,1,1,2)      127: (1,1,1,1,1,1,1)
   16: (5)           63: (1,1,1,1,1,1)    128: (8)
   20: (2,3)         64: (7)              136: (4,4)
   24: (1,4)         72: (3,4)            144: (3,5)
   26: (1,2,2)       80: (2,5)            160: (2,6)
   28: (1,1,3)       84: (2,2,3)          164: (2,3,3)
   30: (1,1,1,2)     96: (1,6)            168: (2,2,4)

The aperiodic case is A334267.
Compositions of this type are counted by A334271.
Normal sequences of this type are counted by A334272.
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, A334266, 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[stc[#]]&&coneckQ[Reverse[stc[#]]]&]

A334297 Length of the Lyndon factorization of the reversed n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2

Offset: 0

Gus Wiseman, Apr 25 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 word is a finite sequence that is prime with respect to the Lyndon product. 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).

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 12345th composition is (1,7,1,1,3,1), with reverse (1,3,1,1,7,1), with Lyndon factorization ((1),(1,3),(1,1,7)), so a(12345) = 3.

The non-reversed version is A329312.
The version for binary indices is A329313 (also the "co-" version).
Positions of 1's are A334265 (reversed Lyndon words).
Binary Lyndon words are counted by A001037 and ranked by A102659.
Lyndon compositions are counted by A059966 and ranked by A275692.
Normal Lyndon sequences are counted by A060223 (row sums of A296372).
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Aperiodic compositions are A328594.
- Distinct rotations are counted by A333632.
- Lyndon factorizations are counted by A333940.
- Length of co-Lyndon factorization is A334029.
Cf. A027375, A211097, A211100, A328595, A329314, A329317.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,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,#]]&]]]];
Table[Length[lynfac[Reverse[stc[n]]]],{n,0,100}]

A333942 Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 5, 7, 9, 11, 7, 11, 11, 15, 7, 12, 16, 21, 16, 26, 26, 36, 12, 21, 26, 36, 21, 36, 36, 52, 11, 19, 29, 38, 31, 52, 52, 74, 29, 52, 66, 92, 52, 92, 92, 135, 19, 38, 52, 74, 52, 92, 92, 135, 38, 74, 92, 135, 74, 135, 135, 203, 15, 30, 47

Offset: 0

Gus Wiseman, Apr 16 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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 a(1) = 1 through a(11) = 11 multiset partitions:
  {1}  {11}    {12}    {111}      {112}      {122}      {123}
       {1}{1}  {1}{2}  {1}{11}    {1}{12}    {1}{22}    {1}{23}
                       {1}{1}{1}  {2}{11}    {2}{12}    {2}{13}
                                  {1}{1}{2}  {1}{2}{2}  {3}{12}
                                                        {1}{2}{3}
  {1111}        {1112}        {1122}        {1123}
  {1}{111}      {1}{112}      {1}{122}      {1}{123}
  {11}{11}      {11}{12}      {11}{22}      {11}{23}
  {1}{1}{11}    {2}{111}      {12}{12}      {12}{13}
  {1}{1}{1}{1}  {1}{1}{12}    {2}{112}      {2}{113}
                {1}{2}{11}    {1}{1}{22}    {3}{112}
                {1}{1}{1}{2}  {1}{2}{12}    {1}{1}{23}
                              {2}{2}{11}    {1}{2}{13}
                              {1}{1}{2}{2}  {1}{3}{12}
                                            {2}{3}{11}
                                            {1}{1}{2}{3}

The described multiset has A000120 distinct parts.
The sum of the described multiset is A029931.
Multisets of compositions are A034691.
The described multiset is a row of A095684.
Combinatory separations of normal multisets are A269134.
The product of the described multiset is A284001.
The version for prime indices is A318284.
The version counting combinatory separations is A334030.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Length of Lyndon factorization is A329312.
- Dealings are counted by A333939.
- Distinct parts are counted by A334028.
- Length of co-Lyndon factorization is A334029.
Cf. A057335, A065609, A275692, A292884, A318560, A318563, A326774, A333764, A333765, A333940.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[Times@@Prime/@ptnToNorm[stc[n]]]],{n,0,30}]

a(n) = A001055(A057335(n)).

A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

Original entry on oeis.org

0, 12, 160, 193, 195, 198, 204, 216, 240, 2304, 2561, 2563, 2566, 2572, 2584, 2608, 2656, 2752, 2944, 3074, 3077, 3079, 3082, 3085, 3087, 3092, 3097, 3099, 3102, 3112, 3121, 3123, 3126, 3132, 3152, 3169, 3171, 3174, 3180, 3192, 3232, 3265, 3267, 3270, 3276

Offset: 1

Gus Wiseman, Nov 21 2021

nonn

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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The terms and corresponding compositions begin:
       0: ()
      12: (1,3)
     160: (2,6)
     193: (1,6,1)
     195: (1,5,1,1)
     198: (1,4,1,2)
     204: (1,3,1,3)
     216: (1,2,1,4)
     240: (1,1,1,5)
    2304: (3,9)
    2561: (2,9,1)
    2563: (2,8,1,1)
    2566: (2,7,1,2)
    2572: (2,6,1,3)
    2584: (2,5,1,4)

These compositions are counted by A224274 up to 0's.
Except for 0, a subset of A345919.
The positive version is A348614, reverse A349153.
An unordered version is A348617, counted by A001523.
The reverse version is A349155.
A positive unordered version is A349159, counted by A000712 up to 0's.
A000346 = even-length compositions with alt sum != 0, complement A001700.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Necklaces are ranked by A065609, dual A333764, reversed A333943.
- Alternating compositions are ranked by A345167, complement A345168.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Total[stc[#]]==-2*ats[stc[#]]&]

A257739 Numbers n for which A256999(n) > n; numbers that can be made larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

Original entry on oeis.org

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 23, 25, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111

Offset: 1

Antti Karttunen, May 18 2015

Note that A256999(a(n)) is always in A257250.

If we define a co-necklace to be a finite sequence that is lexicographically maximal (not minimal) among all of its cyclic rotations, these are numbers whose binary expansion, without the most significant digit, is not a co-necklace. Numbers whose binary expansion, without the most significant digit, is not a necklace are A329367. - Gus Wiseman, Nov 14 2019

			For n = 5 with binary representation "101" if we rotate other bits than the most significant bit (that is, only the two rightmost digits "01") one step to either direction we get "110" = 6 > 5, so 5 can be made larger by such rotations and thus 5 is included in this sequence.
For n = 6 with binary representation "110" no such rotation will yield a larger number and thus 6 is NOT included in this sequence.
For n = 10 with binary representation "1010" if we rotate other bits than the most significant bit (that is, only the three rightmost digits "010") either one step to the left or two steps to the right we get "1100" = 12 > 10, thus 10 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A257250.
Cf. A080541, A080542, A256999.
Numbers whose binary expansion is a necklace are A275692.
Numbers whose binary expansion is a co-necklace are A065609.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose non-msb expansion is a co-necklace are A257250.
Numbers whose non-msb expansion is a necklace are A328668.
Numbers whose reversed non-msb expansion is a necklace are A328607.
Numbers whose non-msb expansion is not a necklace are A329367.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A001037, A003714, A014081, A121016, A164707, A328594, A328596.

reckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[2,100],!reckQ[Rest[IntegerDigits[#,2]]]&] (* Gus Wiseman, Nov 14 2019 *)

A329367 Numbers whose binary expansion, without the most significant digit, is not a necklace.

Original entry on oeis.org

6, 10, 12, 13, 14, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 34, 36, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 83, 84, 86, 88, 89, 90, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Nov 15 2019

nonn

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their binary expansions begins:
   6: (1,1,0)
  10: (1,0,1,0)
  12: (1,1,0,0)
  13: (1,1,0,1)
  14: (1,1,1,0)
  18: (1,0,0,1,0)
  20: (1,0,1,0,0)
  22: (1,0,1,1,0)
  24: (1,1,0,0,0)
  25: (1,1,0,0,1)
  26: (1,1,0,1,0)
  27: (1,1,0,1,1)
  28: (1,1,1,0,0)
  29: (1,1,1,0,1)
  30: (1,1,1,1,0)
  34: (1,0,0,0,1,0)
  36: (1,0,0,1,0,0)
  38: (1,0,0,1,1,0)
  40: (1,0,1,0,0,0)
  41: (1,0,1,0,0,1)

The complement is A328668.
The version involving all digits is A062289.
The reverse version is A328607.
Binary necklaces are A000031.
Necklace compositions are A008965.
Numbers whose binary expansion is a necklace are A275692.
Numbers whose reversed binary expansion is a necklace are A328595.
Cf. A000120, A001037, A003714, A065609, A069010, A257739, A328596.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[2,100],!neckQ[Rest[IntegerDigits[#,2]]]&]

cp's OEIS Frontend

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

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A334274 Numbers k such that the k-th composition in standard order is both a necklace and a reversed co-necklace.

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A334297 Length of the Lyndon factorization of the reversed n-th composition in standard order.

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A333942 Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.

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A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

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A257739 Numbers n for which A256999(n) > n; numbers that can be made larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

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A329367 Numbers whose binary expansion, without the most significant digit, is not a necklace.

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