A211098 Length of largest (i.e., leftmost) Lyndon word in Lyndon factorization of binary vectors of lengths 1, 2, 3, ...

1, 1, 1, 2, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 1, 4, 3, 4, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 5, 3, 5, 4, 5, 2, 2, 2, 5, 3, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 6, 4, 6, 5, 6, 3, 3, 5, 6, 4, 6, 5, 6, 2, 2, 2, 2, 2, 2, 5, 6, 3, 3, 3, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,4

Comments

Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). Here we look at the Lyndon factorizations of the binary vectors 0,1, 00,01,10,11, 000,001,010,011,100,101,110,111, 0000,...

See A211097, A211099, A211100 for further information, including Maple code.

The smallest (or rightmost) factors are given by A211095 and A211096, offset by 2.

References

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.

G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42

Links

Table of n, a(n) for n=1..121.

N. J. A. Sloane, Maple programs for A211097 etc.

Example

Here are the Lyndon factorizations of the first few binary vectors:
.0.
.1.
.0.0.
.01.
.1.0.
.1.1.
.0.0.0.
.001.
.01.0.
.011.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.0.0.0.0.
...

Crossrefs

Cf. A001037 (number of Lyndon words of length m); A102659 (list thereof), A211100.

Cf. A211095-A211099.

Sequence in context: A358172 A344058 A134431 * A070879 A125644 A048821

Adjacent sequences: A211095 A211096 A211097 * A211099 A211100 A211101

Keyword

nonn

Author

N. J. A. Sloane, Apr 01 2012

Status

approved