A329398 -id:A329398 - cp's OEIS frontend

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

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

Offset: 1

Gus Wiseman, Nov 13 2019

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).

			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)

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.

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

Gus Wiseman, Nov 13 2019

nonn

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

			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)

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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.

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}]

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

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

a(19)-a(25) from Robert Price, Jun 20 2021

Terms a(26) and beyond from Andrew Howroyd, Feb 03 2022

A335375 Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.

Original entry on oeis.org

45, 54, 77, 89, 91, 93, 102, 108, 109, 110, 118, 141, 153, 155, 157, 166, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189, 198, 204, 205, 206, 214, 216, 217, 218, 219, 220, 221, 222, 230, 236, 237, 238, 246, 269, 281, 283, 285, 297, 301, 305, 306, 307, 309

Offset: 1

Gus Wiseman, Jun 04 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. It is co-unimodal if its negation is unimodal.

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 together with the corresponding compositions begins:
   45: (2,1,2,1)
   54: (1,2,1,2)
   77: (3,1,2,1)
   89: (2,1,3,1)
   91: (2,1,2,1,1)
   93: (2,1,1,2,1)
  102: (1,3,1,2)
  108: (1,2,1,3)
  109: (1,2,1,2,1)
  110: (1,2,1,1,2)
  118: (1,1,2,1,2)
  141: (4,1,2,1)
  153: (3,1,3,1)
  155: (3,1,2,1,1)
  157: (3,1,1,2,1)
  166: (2,3,1,2)
  173: (2,2,1,2,1)
  177: (2,1,4,1)
  178: (2,1,3,2)
  179: (2,1,3,1,1)

Non-unimodal compositions are ranked by A335373.
Non-co-unimodal compositions are ranked by A335374.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Co-unimodal compositions are A332578.
Numbers with non-co-unimodal unsorted prime signature are A332642.
Non-co-unimodal compositions are A332669.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!unimodQ[stc[#]]&&!unimodQ[-stc[#]]&]

A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

Original entry on oeis.org

13, 22, 25, 27, 29, 38, 41, 44, 45, 46, 49, 50, 51, 53, 54, 55, 57, 59, 61, 70, 76, 77, 78, 81, 82, 83, 86, 88, 89, 90, 91, 92, 93, 94, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 134, 140, 141, 142

Offset: 1

Gus Wiseman, Sep 18 2024

nonn

			The terms and corresponding compositions begin:
  13: (1,2,1)
  22: (2,1,2)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  38: (3,1,2)
  41: (2,3,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)

Eric Weisstein's World of Mathematics, Unimodal Sequence.

The version for run-lengths of compositions is A332833.
Compositions of this type are counted by A332834, complement maybe A329398.
A001523 counts unimodal compositions, ranks too dense.
A011782 counts compositions.
A114994 ranks weakly decreasing compositions, complement A335485.
A115981 counts non-unimodal compositions, ranked by A335373.
A225620 ranks weakly increasing compositions, complement A335486.
A238130, A238279, A333755 count compositions by number of runs.
A332835 counts compositions with weakly incr. or weakly decr. run-lengths.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564.
- Ranks of constant compositions are A272919.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
Cf. A001511, A002051, A065120, A329744, A332745, A332836, A332870, A333218.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!LessEqual@@stc[#]&&!GreaterEqual@@stc[#]&]

Intersection of A335485 and A335486.

cp's OEIS Frontend

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

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

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A335375 Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.

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A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

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