cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A329358 Numbers whose binary expansion has Lyndon and co-Lyndon factorizations of equal lengths.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, 45, 51, 63, 65, 73, 74, 83, 85, 86, 89, 93, 99, 107, 119, 127, 129, 138, 150, 153, 163, 165, 174, 177, 185, 189, 195, 203, 205, 219, 231, 255, 257, 266, 273, 274, 278, 291, 294, 297, 302, 305, 310, 313, 323, 325, 333, 341
Offset: 1

Views

Author

Gus Wiseman, Nov 15 2019

Keywords

Comments

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).
Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).

Examples

			The binary expansions of the initial terms together with their Lyndon and co-Lyndon factorizations:
   1:       (1) =                (1) = (1)
   3:      (11) =             (1)(1) = (1)(1)
   5:     (101) =            (1)(01) = (10)(1)
   7:     (111) =          (1)(1)(1) = (1)(1)(1)
   9:    (1001) =           (1)(001) = (100)(1)
  15:    (1111) =       (1)(1)(1)(1) = (1)(1)(1)(1)
  17:   (10001) =          (1)(0001) = (1000)(1)
  21:   (10101) =        (1)(01)(01) = (10)(10)(1)
  27:   (11011) =        (1)(1)(011) = (110)(1)(1)
  31:   (11111) =    (1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)
  33:  (100001) =         (1)(00001) = (10000)(1)
  45:  (101101) =       (1)(011)(01) = (10)(110)(1)
  51:  (110011) =       (1)(1)(0011) = (1100)(1)(1)
  63:  (111111) = (1)(1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)(1)
  65: (1000001) =        (1)(000001) = (100000)(1)
  73: (1001001) =      (1)(001)(001) = (100)(100)(1)
  74: (1001010) =      (1)(00101)(0) = (100)(10)(10)
  83: (1010011) =      (1)(01)(0011) = (10100)(1)(1)

Crossrefs

The version counting compositions is A329394.
The version ignoring the most significant digit is A329395.
Binary Lyndon/co-Lyndon words are counted by A001037.
Lyndon/co-Lyndon compositions are counted by A059966.
Lyndon compositions whose reverse is not co-Lyndon are A329324.
Binary Lyndon/co-Lyndon words are constructed by A102659 and A329318.
Cf. A060223, A211100, A275692, A328596, A329312, A329313, A329326, A329398.

Programs

  • Mathematica
    lynQ[q_]:=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,#]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Select[Range[100],Length[lynfac[IntegerDigits[#,2]]]==Length[colynfac[IntegerDigits[#,2]]]&]

Formula

A211100(a(n)) = A329312(a(n)).

A329394 Number of compositions of n whose Lyndon and co-Lyndon factorizations both have the same length.

Original entry on oeis.org

1, 2, 2, 4, 4, 10, 13, 28, 46, 99, 175, 359, 672, 1358, 2627, 5238, 10262, 20438, 40320, 80137
Offset: 1

Views

Author

Gus Wiseman, Nov 13 2019

Keywords

Comments

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).
Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).

Examples

			The a(1) = 1 through a(7) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (131)    (33)      (151)
                    (121)   (212)    (141)     (214)
                    (1111)  (11111)  (213)     (232)
                                     (222)     (241)
                                     (231)     (313)
                                     (1221)    (1312)
                                     (2112)    (1321)
                                     (11211)   (2113)
                                     (111111)  (11311)
                                               (12121)
                                               (21112)
                                               (1111111)

Crossrefs

Lyndon and co-Lyndon compositions are (both) counted by A059966.
Lyndon compositions that are not weakly increasing are A329141.
Lyndon compositions of n whose reverse is not co-Lyndon are A329324.
Cf. A000740, A001037, A008965, A060223, A102659, A211100, A275692, A328596, A329312, A329318, A329395, A329398.

Programs

  • Mathematica
    lynQ[q_]:=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,#]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[lynfac[#]]==Length[colynfac[#]]&]],{n,10}]

A329397 Number of compositions of n whose Lyndon factorization is uniform.

Original entry on oeis.org

1, 2, 4, 7, 12, 20, 33, 55, 92, 156, 267, 466, 822, 1473, 2668, 4886, 9021, 16786, 31413, 59101, 111654, 211722, 402697, 768025, 1468170, 2812471, 5397602, 10376418, 19978238, 38519537, 74365161, 143742338, 278156642, 538831403, 1044830113, 2027879831
Offset: 1

Views

Author

Gus Wiseman, Nov 13 2019

Keywords

Comments

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).
A sequence of words is uniform if they all have the same length.

Examples

			The a(1) = 1 through a(6) = 20 Lyndon factorizations:
  (1)  (2)     (3)        (4)           (5)              (6)
       (1)(1)  (12)       (13)          (14)             (15)
               (2)(1)     (112)         (23)             (24)
               (1)(1)(1)  (2)(2)        (113)            (114)
                          (3)(1)        (122)            (123)
                          (2)(1)(1)     (1112)           (132)
                          (1)(1)(1)(1)  (3)(2)           (1113)
                                        (4)(1)           (1122)
                                        (2)(2)(1)        (3)(3)
                                        (3)(1)(1)        (4)(2)
                                        (2)(1)(1)(1)     (5)(1)
                                        (1)(1)(1)(1)(1)  (11112)
                                                         (12)(12)
                                                         (2)(2)(2)
                                                         (3)(2)(1)
                                                         (4)(1)(1)
                                                         (2)(2)(1)(1)
                                                         (3)(1)(1)(1)
                                                         (2)(1)(1)(1)(1)
                                                         (1)(1)(1)(1)(1)(1)

Links

Crossrefs

Lyndon and co-Lyndon compositions are (both) counted by A059966.
Lyndon compositions that are not weakly increasing are A329141.
Lyndon compositions whose reverse is not co-Lyndon are A329324.
Cf. A000740, A001037, A008965, A060223, A102659, A211100, A275692, A328596, A329312, A329318, A329395, A329396, A329398, A329399.

Programs

  • Mathematica
    lynQ[q_]:=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[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Length/@lynfac[#]&]],{n,10}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
B(n,k) = {sumdiv(n, d, moebius(d)/(1-x^d)^(n/d) + O(x*x^k))/n}
seq(n) = {sum(d=1, n-1, my(v=Vec(B(d,n-d),-n)); EulerT(v))} \\ Andrew Howroyd, Feb 03 2022

Formula

G.f.: Sum_{r>=1} (exp(Sum_{k>=1} B(r, x^k)/k) - 1) where B(r, x) = (Sum_{d|r} mu(d)/(1 - x^d)^(r/d))*x^r/r. - Andrew Howroyd, Feb 03 2022

Extensions

a(19)-a(25) from Robert Price, Jun 20 2021
Terms a(26) and beyond from Andrew Howroyd, Feb 03 2022
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