© 2003 V. Keränen

Some structures in  g98

The underlined factors are semi-palindromes.
The longest of them (factor55 - explained below) is of length = 55.

g98a = "abc acdcbcdca db dcbdbabcbdc acbabdbabca bda
dcdadbdcbdbabdbcbacbcdbabdc d bdcacdbcbacbcdcacdcbdcdadbd cbca"

factor55 = w d (mir(w) /. {b ↔ c})

The structure of  factor55  is easier to see from here:

g98  and  g85  compared:

Trying to construct a-2-free strings over  4  letters in some other way

A trial: every second letter is the same 5 times in succession

You cannot continue this string of length 80.  (ok with  aba  but that's all)

Another example:  Use  {a,c}  in odd places and  {b,d}  in even places. How long can you continue?

Answer. This string is the longest possible:

The same string in a readily comprehensible form:

You cannot continue this string either!  Add any letter from {a,b,c,d} and it always creates an abelian square.

So there are long (safe-looking) a-2-free strings that are forbidden as factors in still longer a-2-free strings over  4  letters!

Created by Mathematica  (November 2, 2003)