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A329362 Length of the co-Lyndon factorization of the first n terms of A000002. 9
0, 1, 2, 3, 2, 2, 3, 2, 3, 4, 3, 4, 5, 4, 3, 4, 3, 3, 4, 5, 4, 5, 3, 3, 4, 3, 4, 5, 4, 3, 4, 3, 3, 4, 3, 4, 5, 4, 5, 6, 5, 4, 5, 4, 5, 6, 5, 6, 4, 4, 5, 4, 4, 5, 6, 5, 6, 7, 6, 5, 6, 5, 6, 7, 6, 7, 8, 7, 6, 7, 6, 5, 6, 5, 6, 7, 6, 7, 5, 5, 6, 7, 6, 7, 8, 7, 6, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

LINKS

Table of n, a(n) for n=0..87.

EXAMPLE

The co-Lyndon factorizations of the initial terms of A000002:

                      () = 0

                     (1) = (1)

                    (12) = (1)(2)

                   (122) = (1)(2)(2)

                  (1221) = (1)(221)

                 (12211) = (1)(2211)

                (122112) = (1)(2211)(2)

               (1221121) = (1)(221121)

              (12211212) = (1)(221121)(2)

             (122112122) = (1)(221121)(2)(2)

            (1221121221) = (1)(221121)(221)

           (12211212212) = (1)(221121)(221)(2)

          (122112122122) = (1)(221121)(221)(2)(2)

         (1221121221221) = (1)(221121)(221)(221)

        (12211212212211) = (1)(221121)(2212211)

       (122112122122112) = (1)(221121)(2212211)(2)

      (1221121221221121) = (1)(221121)(221221121)

     (12211212212211211) = (1)(221121)(2212211211)

    (122112122122112112) = (1)(221121)(2212211211)(2)

   (1221121221221121122) = (1)(221121)(2212211211)(2)(2)

  (12211212212211211221) = (1)(221121)(2212211211)(221)

MATHEMATICA

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]:=If[n==0, {}, Nest[kolagrow, {1}, n-1]];

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[colynfac[kol[n]]], {n, 0, 100}]

CROSSREFS

The non-"co" version is A296658.

Cf. A000002, A001037, A060223, A088568, A102659, A275692, A329312, A329315, A329317, A329318.

Sequence in context: A182006 A085239 A242872 * A241604 A282900 A126014

Adjacent sequences:  A329359 A329360 A329361 * A329363 A329364 A329365

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 12 2019

STATUS

approved

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Last modified October 28 07:11 EDT 2021. Contains 348320 sequences. (Running on oeis4.)