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From squares in words to squares in permutations Sergey Kitaev Reykjavík University

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This talk is related to combinatorics on words, a very powerful discipline. 2111=-12+ twoeleven = twelve For example, imaging you have to prove the following identity. Proving it is almost trivial using combinatorics on words!

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A word is a finite or infinite sequence of symbols (letters) taken from a (usually finite) set called alphabet. A factor of a word w is a block of consecutive letters in w. Example: w = abca has 9 distinct factors {a,b,c,ab,bc,ca,abc,bca,abca} A square is a word of the form XX. Example: abcabc, aa, aaaa, baba are squares, whereas abcbca, ab, bab are not squares

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To avoid squares means not to contain them as factors. Example: abaac, 2009, 0101011 do not avoid squares, whereas ABA, eFGFeG, 234 avoid squares. Squares (or any set of prohibited words) are avoidable if there exists an infinite sequence avoiding them. avoidablenon-avoidable 1 letter+ 2 letters+ 3 letters+

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Axel Thue (1906) a abc b ac c b Iterate the following morphism to obtain a square-free sequence abcacbabcbac........ a abcab b acabcb c acbcacb Other morphisms doing the same job: (Thue, 1912) a abcab b abcbac c abcacbc (Thue, 1912) a abcbacbcabcba b bcacbacabcacb c cabacbabcabac (Leech, 1957)

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Axel Thue (1906) a abc b ac c b Iterate the following morphism to obtain a square-free sequence abcacbabcbac........ 1 123...(n-1)n Other morphisms doing the same job: (Arshon, 1937) 2 234...n1 n n12...(n-1)(n-2)... 1 n(n-1)...321 2 1n...432 n (n-2)(n-1)...21n... odd positions even positions

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How many square-free words of length n do we have over k letter alphabet? A tough question! For k=3 the asymptotic answer to the question is 3 cn(1-e ) n where 1.30173...< c <1.30178... and e → when n →. ∞ 0 n Kolpakov, 2006Ochem, Reix, 2006

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Clearly, any permutation is a square-free word. We use one line notation while talking on permutations. Examples of permutations: 25341, 321, 123456, etc. However, we define squares in permutations differently. We say that, e.g., 4257 forms the pattern 2134. A permutation is square-free if no two consecutive factors of length at least 2 are equal as patterns. Example: 246153 contains the square 4615; 246513 is square-free. in pattern form 12-12

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.....

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Question 1: Is the number of square-free permutations finite or infinite? Question 2: If it is infinite, then how does the number of n-permutations grow with growing n? We answer both questions above using a “hot dog”-like construction. An important structural observation:... A B C D E F G H... a segment in a square-free permutation Schematically any square-free permutation looks like this:

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Any permutation! Any square-free permutation! 21 3 1 4 5 2 2 1

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21 3 1 4 5 2 2 1

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21 5 3 6 7 4 2 1

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21 5 3 6 7 4 9 8 Resulting square-free permutation is 523861794.

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Asymptotic enumeartion n/2 n/4

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Asymptotic enumeartion n/2 n/4 coincides with the asymptotics for n!

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Asymptotic enumeartion n/2 n/4

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Directions of further reseach Improving upper bound Connection to permutation patterns avoidance theory: to have a shape above, a permutation must avoid 12 consecutive patterns: 1234, 4321, 2143, 3412, 3142, 2413, 4231, 1324, 4132, 2314, 1423, and 3241.

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Directions of further reseach Improving lower bound 21 5 3 6 7 4 9 8

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Directions of further reseach Improving lower bound 21 5 3 6 7 4 9 8 1 2 3 Resulting square-free permutation is 512863794.

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Thank you for your attention!

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Department of Statistics University of Rajshahi, Bangladesh

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