# Abelian Patterns

Given a pattern (i.e., a word)
p = p1p2 ... pn
we say that a word w encounters p in the Abelian sense if w contains a factor P1P2 ... Pn where word Pi is an anagram of word Pj whenever pi = pj. Otherwise w avoids p in the Abelian sense and is Abelian p-free. As an example, the word
reproportioners = re pro por tion er s
encounters abbca in the Abelian sense.

We say that pattern p is avoidable in the Abelian sense if for some fixed finite alphabet Σ there are arbitrarily long words over Σ which avoid p. Equivalently, there is an infinite p-free sequence over Σ. We say that p is k-avoidable in the Abelian sense there is an infinite Abelian p-free sequence over {1, 2, 3, ..., k}. Keränen showed that xx is Abelian 4-avoidable; Dekking showed that xxx is Abelian 3-avoidable and xxxx is Abelian 2-avoidable. Avoiding patterns in the Abelian sense generalizes an existing theory of pattern avoidance [2,3,19]. (We sometimes speak of avoiding patterns "in the ordinary sense".) To encounter p in the "ordinary sense", we modify the definition above to insist that Pi = Pj whenever pi = pj. It is decidable whether a pattern p is avoidable in this sense, while the corresponding decision problem for k-avoidability is open. The word wΔ given in [2] is 4-avoidable, but not 3-avoidable. In [8] a pattern is given which is 5-avoidable, but not 4-avoidable.

### Open Problem: Is there a pattern p which is 6-avoidable but not 5-avoidable?

The complexity of deciding avoidability remains open, although unsubstantiated claims have been made that deciding avoidability is NP-complete, or even exponential.

### Open Problem: Given pattern p, what is the complexity of deciding whether p is avoidable?

In the case of words avoiding Abelian patterns, even less is known. In [7], Abelian fractional powers are defined; a word xyx' where x' is an anagram of x is an Abelian k-power where k=|xyx'|/|xy|. It is proved that for fixed 1 < k < 2 there exists an alphabet size nk such that an infinite word exists over {1, 2, ..., nk } not containing any Abelian k-power. In [11], this is used to prove that certain patterns are avoidable in the Abelian sense. The pattern
is shown to be avoidable, but not Abelian avoidable.

### Open Problem: Which patterns are Abelian avoidable?

Here are some quite specific subproblems left over from [11]:

### Open Problem: Which of the following patterns 01020312 01020321 01021303 01023031 010203013 010213020 is avoidable in the Abelian sense?

In [12, 13], a start is made on a theory of Abelian k-avoidablity. (See this link for some slides.) It is shown that binary patterns of length 119 or more are Abelian 2-avoidable. This leaves a large but approachable project:

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