From squares in words to squares in permutations

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

2. This talk is related to combinatorics on words, a very powerful discipline. For example, imaging you have to prove the following identity. Proving it is almost trivial using combinatorics on words! 2 + 11 - 1 = 12 two eleven twelve =

3. A wordis a finite or infinite sequence of symbols (letters) taken from a (usually finite) set called alphabet. A factorof 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 squareis a word of the form XX. Example: abcabc, aa, aaaa, baba are squares, whereas abcbca, ab, bab are not squares

4. To avoidsquares 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.

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

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

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

8. We use one line notation while talking on permutations. Examples of permutations: 25341, 321, 123456, etc. Clearly, any permutation is a square-free word. 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

9. .....

10. 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:

11. Any square-free permutation! Any permutation! 2 1 3 4 2 1 5 2 1 Any permutation!

12. 2 1 3 4 2 1 5 2 1

13. 2 1 5 6 4 3 7 2 1

14. 9 8 5 6 4 3 7 2 1 Resulting square-free permutation is 523861794.

15. Asymptotic enumeartion n/4 n/2 n/4

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

17. Asymptotic enumeartion n/4 n/2 n/4

18. 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.

19. Directions of further reseach Improving lower bound 9 8 5 6 4 3 7 2 1

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

21. Thank you for your attention!