# [seqfan] Re: Seqfan Digest, Vol 36, Issue 1

franktaw at netscape.net franktaw at netscape.net
Fri Sep 9 05:45:09 CEST 2011

```See comments inline.

-----Original Message-----
>From: Ed Jeffery <ed.jeffery at yahoo.com>

>Seqfans,

>Regarding A194923, what is the significance of writing "abelian
square-free"
>rather than "square-free?"

This is explained in the first comment to A194923. A square is two
subsequences that contain the same multiset of symbols: the same
symbols, including frequencies, but not necessarily in the same order.

>Also, in the links for A194923Keränen writes:

>"It is easily seen that abelian squares cannot be avoided over a three
letter
>alphabet. Indeed, in this alphabet, each word of length 8 contains an
abelian
>square..."

>If the above assertion is true, since we are dealing with ternary
abelian
>square-free words, doesn't it mean that the row of A194923 presumably
containing
>words of length 8 should be written as {{}}, thus disrupting the
integer
>sequence?

Note that A194923 has the "fini" keyword, indicating that it is a
finite sequence. The example and the b-file each contain the entire
sequence.

>On the other hand, aren't e.g. 01202120, 01020121and 01210201
>square-free, or are they square-free but not abelian square-free?
>the difference.

See above. E.g., in 01202120, we can group this as (012)(021)20 and the
(012)(021) is an Abelian square.

>Ed Jeffery