A296659 Length of the final word in the standard Lyndon word factorization of the first n terms of A000002.

1, 2, 3, 1, 1, 3, 1, 5, 6, 1, 8, 9, 1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 1, 3, 1, 14, 15, 1, 1, 3, 1, 1, 3, 1, 8, 9, 1, 11, 12, 1, 1, 3, 1, 17, 18, 1, 20, 1, 1, 3, 1, 1, 3, 27, 1, 29, 30, 1, 1, 3, 1, 35, 36, 1, 38, 39, 1, 1, 3, 1, 1, 3, 1, 8, 9, 1, 11, 1, 1, 3, 15, 1

Offset

1,2

Links

Table of n, a(n) for n=1..82.

Frédérique Bassino, Julien Clement, and Cyril Nicaud, The standard factorization of Lyndon words: an average point of view, Discrete Mathematics, 290-1, (2005), 1-25.

Example

The sequence of final words begins: 1, 12, 122, 1, 1, 112, 1, 11212, 112122, 1, 11212212, 112122122, 1, 1, 112, 1, 1, 112, 1121122, 1, 112112212, 1, 1, 112, 1, 11211221211212, 112112212112122, 1, 1, 112.

Mathematica

LyndonQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And]&&Array[RotateRight[q, #]&, Length[q], 1, UnsameQ];
qit[q_]:=If[#===Length[q], {q}, Prepend[qit[Drop[q, #]], Take[q, #]]]&[Max@@Select[Range[Length[q]], LyndonQ[Take[q, #]]&]];
kolagrow[q_]:=If[Length[q]<2, Take[{1, 2}, Length[q]+1], Append[q, Switch[{q[[Length[Split[q]]]], Part[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]]];
Table[Length[Last[qit[Nest[kolagrow, 1, n]]]], {n, 150}]

Crossrefs

Cf. A000002, A027375, A088568, A102659, A228369, A281013, A288605, A296372, A296657, A296658.

Sequence in context: A356152 A317627 A271566 * A270823 A067627 A077233

Adjacent sequences: A296656 A296657 A296658 * A296660 A296661 A296662

Keyword

nonn

Author

Gus Wiseman, Dec 18 2017

Status

approved