A328594 -id:A328594 - cp's OEIS frontend

A333939 Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order.

1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 5, 4, 5, 1, 2, 2, 4, 2, 4, 5, 7, 2, 5, 4, 10, 4, 10, 7, 7, 1, 2, 2, 4, 2, 5, 5, 7, 2, 5, 3, 9, 5, 13, 11, 12, 2, 5, 5, 10, 5, 11, 13, 18, 4, 10, 9, 20, 7, 18, 12, 11, 1, 2, 2, 4, 2, 5, 5, 7, 2, 4, 4, 11, 5, 14, 11, 12, 2

Offset: 0

Gus Wiseman, Apr 15 2020

nonn

Number of ways to deal out the k-th composition in standard order to form a multiset of hands.

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 dealings for n = 1, 3, 7, 11, 13, 23, 43:
  (1)  (11)    (111)      (211)      (121)      (2111)        (2211)
       (1)(1)  (1)(11)    (1)(21)    (1)(12)    (11)(21)      (11)(22)
               (1)(1)(1)  (2)(11)    (1)(21)    (1)(211)      (1)(221)
                          (1)(1)(2)  (2)(11)    (2)(111)      (21)(21)
                                     (1)(1)(2)  (1)(1)(21)    (2)(211)
                                                (1)(2)(11)    (1)(1)(22)
                                                (1)(1)(1)(2)  (1)(2)(21)
                                                              (2)(2)(11)
                                                              (1)(1)(2)(2)

Multisets of compositions are counted by A034691.
Combinatory separations of normal multisets are counted by A269134.
Dealings with total sum n are counted by A292884.
Length of co-Lyndon factorization of binary expansion is A329312.
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 words are A275692.
- Co-Lyndon words are A326774.
- Aperiodic compositions are A328594.
- Length of Lyndon factorization is A329312.
- Distinct rotations are counted by A333632.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Length of co-Lyndon factorization is A334029.
- Combinatory separations are A334030.
Cf. A000031, A000740, A001037, A008965, A027375, A059966, A060223, A211100, A328595, A328596, A333764, A333943.

nn=100;
comps[0]:={{}};comps[n_]:=Join@@Table[Prepend[#,i]&/@comps[n-i],{i,n}];
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[dealings[stc[n]]],{n,0,nn}]

For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A292884(n).

A329134 Numbers whose differences of prime indices are a periodic word.

Original entry on oeis.org

8, 16, 27, 30, 32, 64, 81, 105, 110, 125, 128, 180, 210, 238, 243, 256, 273, 343, 385, 450, 506, 512, 625, 627, 729, 806, 935, 1001, 1024, 1080, 1100, 1131, 1155, 1331, 1394, 1495, 1575, 1729, 1786, 1870, 1887, 2048, 2187, 2197, 2310, 2401, 2431, 2451, 2635

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A sequence is periodic if its cyclic rotations are not all different.

			The sequence of terms together with their differences of prime indices begins:
     8: (0,0)
    16: (0,0,0)
    27: (0,0)
    30: (1,1)
    32: (0,0,0,0)
    64: (0,0,0,0,0)
    81: (0,0,0)
   105: (1,1)
   110: (2,2)
   125: (0,0)
   128: (0,0,0,0,0,0)
   180: (0,1,0,1)
   210: (1,1,1)
   238: (3,3)
   243: (0,0,0,0)
   256: (0,0,0,0,0,0,0)
   273: (2,2)
   343: (0,0)
   385: (1,1)
   450: (1,0,1,0)

Complement of A329135.
These are the Heinz numbers of the partitions counted by A329144.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Cf. A000740, A027375, A056239, A112798, A124010, A328594, A329132, A329139.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[10000],!aperQ[Differences[primeMS[#]]]&]

A329135 Numbers whose differences of prime indices are an aperiodic word.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A sequence is aperiodic if its cyclic rotations are all different.

			The sequence of terms together with their differences of prime indices begins:
    1: ()
    2: ()
    3: ()
    4: (0)
    5: ()
    6: (1)
    7: ()
    9: (0)
   10: (2)
   11: ()
   12: (0,1)
   13: ()
   14: (3)
   15: (1)
   17: ()
   18: (1,0)
   19: ()
   20: (0,2)
   21: (2)
   22: (4)

Complement of A329134.
These are the Heinz numbers of the partitions counted by A329137.
Aperiodic compositions are A000740.
Aperiodic binary words are A027375.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A124010, A152061, A329133, A329136, A329140.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Differences[primeMS[#]]]&]

A329132 Numbers whose augmented differences of prime indices are a periodic sequence.

Original entry on oeis.org

4, 8, 15, 16, 32, 55, 64, 90, 105, 119, 128, 225, 253, 256, 403, 512, 540, 550, 697, 893, 935, 1024, 1155, 1350, 1357, 1666, 1943, 2048, 2263, 3025, 3071, 3150, 3240, 3375, 3451, 3927, 3977, 4096, 4429, 5123, 5500, 5566, 6731, 7735, 8083, 8100, 8192, 9089

Offset: 1

Gus Wiseman, Nov 06 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A sequence is periodic if its cyclic rotations are not all different.

			The sequence of terms together with their augmented differences of prime indices begins:
     4: (1,1)
     8: (1,1,1)
    15: (2,2)
    16: (1,1,1,1)
    32: (1,1,1,1,1)
    55: (3,3)
    64: (1,1,1,1,1,1)
    90: (2,1,2,1)
   105: (2,2,2)
   119: (4,4)
   128: (1,1,1,1,1,1,1)
   225: (1,2,1,2)
   253: (5,5)
   256: (1,1,1,1,1,1,1,1)
   403: (6,6)
   512: (1,1,1,1,1,1,1,1,1)
   540: (2,1,1,2,1,1)
   550: (3,1,3,1)
   697: (7,7)
   893: (8,8)

Complement of A329133.
These are the Heinz numbers of the partitions counted by A329143.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Numbers whose differences of prime indices are periodic are A329134.
Cf. A000961, A027375, A056239, A112798, A325356, A325389, A325394, A328594, A329135, A329136, A329139.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
aug[y_]:=Table[If[i

A329136 Number of integer partitions of n whose augmented differences are an aperiodic word.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 10, 14, 19, 28, 40, 53, 75, 99, 131, 172, 226, 294, 380, 488, 617, 787, 996, 1250, 1565, 1953, 2425, 3003, 3705, 4559, 5589, 6836, 8329, 10132, 12292, 14871, 17950, 21629, 25988, 31169, 37306, 44569, 53139, 63247, 75133, 89111, 105515, 124737

Offset: 0

Gus Wiseman, Nov 09 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

A sequence is aperiodic if its cyclic rotations are all different.

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)  (3)    (4)      (5)        (6)          (7)
            (2,1)  (2,2)    (4,1)      (3,3)        (4,3)
                   (3,1)    (2,2,1)    (4,2)        (5,2)
                   (2,1,1)  (3,1,1)    (5,1)        (6,1)
                            (2,1,1,1)  (2,2,2)      (3,2,2)
                                       (3,2,1)      (3,3,1)
                                       (4,1,1)      (4,2,1)
                                       (2,2,1,1)    (5,1,1)
                                       (3,1,1,1)    (2,2,2,1)
                                       (2,1,1,1,1)  (3,2,1,1)
                                                    (4,1,1,1)
                                                    (2,2,1,1,1)
                                                    (3,1,1,1,1)
                                                    (2,1,1,1,1,1)
With augmented differences:
  (1)  (2)  (3)    (4)      (5)        (6)          (7)
            (2,1)  (1,2)    (4,1)      (1,3)        (2,3)
                   (3,1)    (1,2,1)    (3,2)        (4,2)
                   (2,1,1)  (3,1,1)    (5,1)        (6,1)
                            (2,1,1,1)  (1,1,2)      (1,3,1)
                                       (2,2,1)      (2,1,2)
                                       (4,1,1)      (3,2,1)
                                       (1,2,1,1)    (5,1,1)
                                       (3,1,1,1)    (1,1,2,1)
                                       (2,1,1,1,1)  (2,2,1,1)
                                                    (4,1,1,1)
                                                    (1,2,1,1,1)
                                                    (3,1,1,1,1)
                                                    (2,1,1,1,1,1)

The Heinz numbers of these partitions are given by A329133.
The periodic version is A329143.
The non-augmented version is A329137.
Aperiodic binary words are A027375.
Aperiodic compositions are A000740.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose differences of prime indices are aperiodic are A329135.
Numbers whose prime signature is aperiodic are A329139.
Cf. A152061, A325351, A325356, A329132, A329134, A329139, A329140.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
aug[y_]:=Table[If[i

a(n) + A329143(n) = A000041(n).

A334269 Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 16, 23, 40, 62, 110, 169, 302, 492, 856, 1454, 2572, 4428, 7914, 13935, 25036, 44842, 81298, 147149, 268952, 491746, 904594, 1667091, 3085950, 5723367, 10652544, 19865887, 37150314, 69608939, 130723184, 245935633, 463590444, 875306913, 1655451592, 3135613649, 5948011978, 11298215516

Offset: 1

Gus Wiseman, Apr 24 2020

nonn

Also the number of compositions of n that are both a Lyndon word and a reversed co-Lyndon word.
A composition of n is a finite sequence of positive integers summing to n.
A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations. Co-Lyndon is defined similarly, except with strictly greater instead of strictly less.

			The a(1) = 1 through a(7) = 16 compositions:
  (1)  (2)  (3)   (4)    (5)     (6)      (7)
            (21)  (31)   (32)    (42)     (43)
                  (211)  (41)    (51)     (52)
                         (221)   (321)    (61)
                         (311)   (411)    (322)
                         (2111)  (2211)   (331)
                                 (3111)   (421)
                                 (21111)  (511)
                                          (2221)
                                          (3121)
                                          (3211)
                                          (4111)
                                          (21211)
                                          (22111)
                                          (31111)
                                          (211111)

The version for binary expansion is A334267.
Compositions of this type are ranked by A334266.
Normal sequences of this type are counted by A334270.
Necklace compositions of this type are counted by A334271.
Aperiodic compositions are counted by A000740.
Binary Lyndon words are counted by A001037.
Necklace compositions are counted by A008965.
Normal Lyndon words are counted by A060223.
Lyndon compositions are counted by A059966.
All of the following pertain to compositions in standard order (A066099):
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed Lyndon words are A334265.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Length of co-Lyndon factorization is A334029.
- Length of Lyndon factorization of reverse is A334297.
- Length of co-Lyndon factorization of reverse is A329313.
- Lyndon factorizations are counted by A333940.
- Co-Lyndon factorizations are counted by A333765.
- Aperiodic compositions are A328594.
- Distinct rotations are counted by A333632.
Cf. A034691, A065609, A275692, A328596, A329141, A329324, A329326, A334266, A334272, A334273, A334274.

lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],lynQ[Reverse[#]]&&colynQ[#]&]],{n,0,15}]

Offset corrected and a(21)-a(42) from Bert Dobbelaere, Apr 26 2020

A334271 Number of compositions of n that are both a reversed necklace and a co-necklace.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 28, 43, 70, 111, 184, 303, 510, 865, 1482, 2573, 4480, 7915, 14008

Offset: 0

Gus Wiseman, Apr 25 2020

Also the number of compositions of n that are both a necklace and a reversed co-necklace.

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

			The a(1) = 1 through a(6) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (21)   (22)    (32)     (33)
             (111)  (31)    (41)     (42)
                    (211)   (221)    (51)
                    (1111)  (311)    (222)
                            (2111)   (321)
                            (11111)  (411)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (21111)
                                     (111111)

Normal sequences of this type are counted by A334272.
The aperiodic case is A334269.
These compositions are ranked by A334273.
Binary (or reversed binary) necklaces are counted by A000031.
Normal sequences are counted by A000670.
Necklace compositions are counted by A008965.
Lyndon compositions are counted by A059966.
Normal Lyndon words are counted by A060223.
Normal necklaces are counted by A019536.
Normal aperiodic words are counted by A296975.
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. A000740, A001037, A034691, A329138, A329324, A334266, A334267, A334274.

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];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[Reverse[#]]&&coneckQ[#]&]],{n,0,15}]

A334272 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed necklace and a co-necklace.

Original entry on oeis.org

1, 1, 2, 4, 12, 43, 229, 1506, 12392, 120443

Offset: 0

Gus Wiseman, Apr 25 2020

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

			The a(1) = 1 through a(4) = 12 normal sequences:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)
       (2,1)  (2,1,1)  (2,1,1,1)
              (2,2,1)  (2,1,2,1)
              (3,2,1)  (2,2,1,1)
                       (2,2,2,1)
                       (3,1,2,1)
                       (3,2,1,1)
                       (3,2,2,1)
                       (3,2,3,1)
                       (3,3,2,1)
                       (4,2,3,1)
                       (4,3,2,1)

Dominates A334270 (the aperiodic case).
Compositions of this type are counted by A334271.
These compositions are ranked by A334273 (standard) and A334274 (binary).
Binary (or reversed binary) necklaces are counted by A000031.
Normal sequences are counted by A000670.
Necklace compositions are counted by A008965.
Normal Lyndon words are counted by A060223.
Normal necklaces are counted by A019536.
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.
- Reversed Lyndon co-Lyndon compositions are A334266.
- Aperiodic compositions are A328594.
Cf. A034691, A059966, A296372, A296975, A329138, A334269.

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];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],neckQ[Reverse[#]]&&coneckQ[#]&]],{n,0,8}]

A328607 Numbers whose reversed binary expansion, without the most significant digit, is a necklace.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 26, 28, 30, 31, 32, 48, 52, 56, 58, 60, 62, 63, 64, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 127, 128, 192, 200, 208, 212, 216, 220, 224, 228, 232, 234, 236, 240, 244, 246, 248, 250, 252, 254, 255

Offset: 0

Gus Wiseman, Oct 30 2019

nonn

Offset is 0 to be consistent with A257250.

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 and binary indices begins:
    0:        0 ~ {}
    1:        1 ~ {1}
    2:       10 ~ {2}
    3:       11 ~ {1,2}
    4:      100 ~ {3}
    6:      110 ~ {2,3}
    7:      111 ~ {1,2,3}
    8:     1000 ~ {4}
   12:     1100 ~ {3,4}
   14:     1110 ~ {2,3,4}
   15:     1111 ~ {1,2,3,4}
   16:    10000 ~ {5}
   24:    11000 ~ {4,5}
   26:    11010 ~ {2,4,5}
   28:    11100 ~ {3,4,5}
   30:    11110 ~ {2,3,4,5}
   31:    11111 ~ {1,2,3,4,5}
   32:   100000 ~ {6}
   48:   110000 ~ {5,6}
   52:   110100 ~ {3,5,6}

The dual non-reversed version is A257250.
The dual non-reversed version involving all digits is A065609.
The version involving all digits is A328595.
The non-reversed version is A328668.
Binary necklaces are A000031.
Cf. A000120, A001037, A003714, A008965, A014081, A121016, A164707, A275692, A328594, A328596.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[0,100],#<=1||neckQ[Reverse[Rest[IntegerDigits[#,2]]]]&]

A328668 Numbers whose binary expansion, without the most significant digit, is a necklace.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19, 21, 23, 31, 32, 33, 35, 37, 39, 43, 47, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 85, 87, 91, 95, 127, 128, 129, 131, 133, 135, 137, 139, 141, 143, 147, 149, 151, 155, 157, 159, 171, 175, 183, 191, 255, 256, 257

Offset: 0

Gus Wiseman, Oct 26 2019

nonn

Offset is 0 to be consistent with A257250.

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 and binary indices begins:
   0:       0 ~ {}
   1:       1 ~ {1}
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   4:     100 ~ {3}
   5:     101 ~ {1,3}
   7:     111 ~ {1,2,3}
   8:    1000 ~ {4}
   9:    1001 ~ {1,4}
  11:    1011 ~ {1,2,4}
  15:    1111 ~ {1,2,3,4}
  16:   10000 ~ {5}
  17:   10001 ~ {1,5}
  19:   10011 ~ {1,2,5}
  21:   10101 ~ {1,3,5}
  23:   10111 ~ {1,2,3,5}
  31:   11111 ~ {1,2,3,4,5}
  32:  100000 ~ {6}
  33:  100001 ~ {1,6}
  35:  100011 ~ {1,2,6}

The dual version is A257250.
The version involving all digits, taken in reverse, is A328595.
The reversed version is A328607.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A001037, A003714, A014081, A065609, A121016, A275692, A328594, A328596.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[0,100],#<=1||neckQ[Rest[IntegerDigits[#,2]]]&]

cp's OEIS Frontend

A333939 Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order.

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A329134 Numbers whose differences of prime indices are a periodic word.

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A329135 Numbers whose differences of prime indices are an aperiodic word.

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A329132 Numbers whose augmented differences of prime indices are a periodic sequence.

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A329136 Number of integer partitions of n whose augmented differences are an aperiodic word.

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A334269 Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.

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A334271 Number of compositions of n that are both a reversed necklace and a co-necklace.

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A334272 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed necklace and a co-necklace.

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A328607 Numbers whose reversed binary expansion, without the most significant digit, is a necklace.

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