Equivalence classes of length-changing replacements of size-3 patterns

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1 Equivalence classes of length-changing replacements of size-3 patterns Vahid Fazel-Rezai Mentor: Tanya Khovanova 2013 MIT-PRIMES Conference May 18, 2013 Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

2 Outline 1 Definitions Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

3 Outline 1 Definitions 2 Results β Decreasing Shift Right, Shift Left Drop Only Drop One, Swap with Neighbor Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

4 Outline 1 Definitions 2 Results β Decreasing Shift Right, Shift Left Drop Only Drop One, Swap with Neighbor 3 Future Plans Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

5 Definitions Permutations and Patterns Definition A permutation is a string consisting of 1, 2, 3,..., n. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

6 Definitions Permutations and Patterns Definition A permutation is a string consisting of 1, 2, 3,..., n. Special permutations: n (identity permutation) (empty permutation) Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

7 Definitions Permutations and Patterns Definition A permutation is a string consisting of 1, 2, 3,..., n. Special permutations: n (identity permutation) (empty permutation) Definition Let p be a string of distinct positive integers. A substring of a permutation π order-isomorphic to p is a copy of the pattern p in π. If no such substrings exist, π avoids p. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

8 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

9 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

10 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

11 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

12 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

13 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

14 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

15 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

16 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

17 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

18 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

19 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

20 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

21 Definitions Replacements Definition Let α and β be strings, of equal length, of distinct integers and. Then, σ is the result of a replacement α β on π if σ is obtained by: 1 adding instances of in π as necessary, 2 replacing a copy of the pattern α with β, 3 dropping all instances of, and then 4 relabeling as necessary. Example: Under , behold! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

22 Definitions Equivalence Definition Two permutations π and σ are equivalent (π σ) under α β if σ can be attained through a sequence of α β or β α replacements on π. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

23 Definitions Equivalence Definition Two permutations π and σ are equivalent (π σ) under α β if σ can be attained through a sequence of α β or β α replacements on π. Example: Under , we have Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

24 Our Focus Definitions We are interested in equivalence classes. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

25 Definitions Our Focus We are interested in equivalence classes. We look at replacements of the type 123 β where β has one (total of 18 cases). Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

26 Definitions Our Focus We are interested in equivalence classes. We look at replacements of the type 123 β where β has one (total of 18 cases). We organize these replacements into four categories: Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

27 Definitions Our Focus We are interested in equivalence classes. We look at replacements of the type 123 β where β has one (total of 18 cases). We organize these replacements into four categories: β Decreasing (9 cases) e.g Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

28 Definitions Our Focus We are interested in equivalence classes. We look at replacements of the type 123 β where β has one (total of 18 cases). We organize these replacements into four categories: β Decreasing (9 cases) e.g Shift Right, Shift Left (2 cases) and Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

29 Definitions Our Focus We are interested in equivalence classes. We look at replacements of the type 123 β where β has one (total of 18 cases). We organize these replacements into four categories: β Decreasing (9 cases) e.g Shift Right, Shift Left (2 cases) and Drop Only (3 cases) e.g Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

30 Definitions Our Focus We are interested in equivalence classes. We look at replacements of the type 123 β where β has one (total of 18 cases). We organize these replacements into four categories: β Decreasing (9 cases) e.g Shift Right, Shift Left (2 cases) and Drop Only (3 cases) e.g Drop, Swap with Neighbor (4 cases) e.g Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

31 Outline Results β Decreasing 1 Definitions 2 Results β Decreasing Shift Right, Shift Left Drop Only Drop One, Swap with Neighbor 3 Future Plans Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

32 Results β Decreasing Two Lemmas Lemma If β is decreasing, then any permutation is equivalent under 123 β to some identity permutation. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

33 Results β Decreasing Two Lemmas Lemma If β is decreasing, then any permutation is equivalent under 123 β to some identity permutation. The above is because a descent can be replaced with an increasing substring. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

34 Results β Decreasing Two Lemmas Lemma If β is decreasing, then any permutation is equivalent under 123 β to some identity permutation. The above is because a descent can be replaced with an increasing substring. Lemma If β is decreasing, all identity permutations of length 4 or greater are equivalent. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

35 Results β Decreasing Finitely Many Classes Theorem If β is decreasing, there are five equivalence classes: { }, {1}, {12}, {123, 21}, {everything else} Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

36 Outline Results Shift Right, Shift Left 1 Definitions 2 Results β Decreasing Shift Right, Shift Left Drop Only Drop One, Swap with Neighbor 3 Future Plans Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

37 Results Shift Right, Shift Left Reverse Identities Isolated We observe the following: All identities of length 2 or greater are equivalent. All permutations of length 3, except 321, are equivalent. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

38 Results Shift Right, Shift Left Reverse Identities Isolated We observe the following: All identities of length 2 or greater are equivalent. All permutations of length 3, except 321, are equivalent. Theorem Under and , each reverse identity is in a distinct class while all other permutations are equivalent. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

39 Outline Results Drop Only 1 Definitions 2 Results β Decreasing Shift Right, Shift Left Drop Only Drop One, Swap with Neighbor 3 Future Plans Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

40 Results Drop Only Shortest Equivalent Permutation Lemma Apply the replacement 123 β as many times as possible (in any order) to some π, and call the result p(π). p(π) is the unique shortest permutation equivalent to π. p(π) avoids 123. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

41 Results Drop Only Shortest Equivalent Permutation Lemma Apply the replacement 123 β as many times as possible (in any order) to some π, and call the result p(π). p(π) is the unique shortest permutation equivalent to π. p(π) avoids 123. Thus, there is a bijection between equivalence classes and permutations avoiding 123: Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

42 Results Drop Only Shortest Equivalent Permutation Lemma Apply the replacement 123 β as many times as possible (in any order) to some π, and call the result p(π). p(π) is the unique shortest permutation equivalent to π. p(π) avoids 123. Thus, there is a bijection between equivalence classes and permutations avoiding 123: Theorem Under drop only replacements, for each σ avoiding 123, there exists a distinct class containing all π with p(π) = σ. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

43 Outline Results Drop One, Swap with Neighbor 1 Definitions 2 Results β Decreasing Shift Right, Shift Left Drop Only Drop One, Swap with Neighbor 3 Future Plans Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

44 Alternative Equivalence Results Drop One, Swap with Neighbor The following lemma allows previous work to be applied here. Lemma There exists some length-preserving replacement under which equivalence implies equivalence under 123 β for each 123 β in this category. For example, equivalence under implies equivalence under Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

45 Results Drop One, Swap with Neighbor Characterizing Classes by Invariants Definition An element of a permutation is called a left-to-right minimum if it has a value smaller than every element to its left. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

46 Results Drop One, Swap with Neighbor Characterizing Classes by Invariants Definition An element of a permutation is called a left-to-right minimum if it has a value smaller than every element to its left. Theorem Two permutations are equivalent under if and only if they have the following in common: number of left-to-right minima, and out of the elements that are not left-to-right minima, leftmost position, and largest value (relative to left-right minima). The other three replacements have similar invariants. Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

47 Future Plans Future Work I plan to continue this research by: characterizing equivalence classes of 132 β replacements considering the case when β contains two generalizing to longer patterns exploring the shortest distance between two permutations examining why some replacements have the same classes Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

48 Acknowledgments Thank You! I would like to thank: my mentor Tanya Khovanova for extremely valuable guidance Prof. James Propp for suggesting the project and continually providing direction PRIMES for allowing this research to happen Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

49 Acknowledgments Thank You! I would like to thank: my mentor Tanya Khovanova for extremely valuable guidance Prof. James Propp for suggesting the project and continually providing direction PRIMES for allowing this research to happen Thank you for listening! Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

50 Acknowledgments Thank You! I would like to thank: my mentor Tanya Khovanova for extremely valuable guidance Prof. James Propp for suggesting the project and continually providing direction PRIMES for allowing this research to happen Thank you for listening! Questions? Vahid Fazel-Rezai Length-Changing Pattern Replacements PRIMES / 18

Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns

Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns Vahid Fazel-Rezai Phillips Exeter Academy Exeter, New Hampshire, U.S.A. vahid fazel@yahoo.com Submitted: Sep