© 2003 V. Keränen

You may copy the image words of g85 and g98 as text from here

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!

How did we find these examples?

More details:

Created by Mathematica (November 2, 2003)