A160675 Duplication root: the maximal number of distinct squarefree words that a word of length n can be reduced to by iterated application of string-rewriting rules uu->u.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5

Offset

1,9

Comments

The growth is bounded by 1/30*110^(n/42) <= DupRoots(n) <= 2^n.

Duplication on strings was originally motivated by the fact that it occurs in DNA strands.

References

J. Dassow, V. Mitrana, and G. Păun, On the regularity of duplication closure. Bulletin of the EATCS, 69 1999, pp. 133-136.

Peter Leupold, Duplication roots, in Developments in Language Theory, T. Harju, J. Karhumski and A. Lepisto, eds., vol. 4588 of Lecture Notes in Computer Science, Springer, 2007, pp. 290-299.

Links

Table of n, a(n) for n=1..25.

Peter Leupold, Reducing Repetitions, Stringology (Aug 2009), pp. 225-236.

Example

The shortest word with ambiguous root (up to reversal and renaming of letter) is
.abcbabcbc
which can be reduced to the words
.abc, abcbc, abcbabc, abcbabcbc
and of these only
.abc, abcbabc
are squarefree.
Witnesses for the value changes from 2 to 4 and from 4 to 5 are
.DUPROOT(abcbabcbcacbca) = (abcbabcacbca, abcbabca, abcacbca, abca).
.DUPROOT(ababcbabcacbabcabacbabcab) = (abcbabcabacbabcab, abcbabcab, abcacbabcab, abcabacbabcab, abcab).
Words with three roots exist, however, the first one is longer than the first one with four roots:
.DUPROOT(babacabacbcabacb) = (bacabacb, bacbcabacb, bacb).

Crossrefs

Sequence in context: A294078 A064133 A295101 * A105674 A130496 A187243

Adjacent sequences: A160672 A160673 A160674 * A160676 A160677 A160678

Keyword

hard,more,nonn,nice

Author

Peter Leupold (leupold(AT)cc.kyoto-su.ac.jp), May 23 2009

Status

approved