Schemes for Pattern-Avoiding Words Lara Pudwell Rutgers University - PowerPoint PPT Presentation

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Schemes for Pattern-Avoiding Words Lara Pudwell Rutgers University Permutation Patterns 2007 Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Other Schemes for Words Summary Outline Introduction/History 1 Pattern Avoidance in Words Previous Work Prefix Schemes for Words 2 Definitions Examples Success Rate Other Schemes for Words 3 Schemes for Monotone Patterns Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Outline Introduction/History 1 Pattern Avoidance in Words Previous Work Prefix Schemes for Words 2 Definitions Examples Success Rate Other Schemes for Words 3 Schemes for Monotone Patterns Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Reduction Given a string of letters p = p 1 ... p n , the reduction of p is the string obtained by replacing the i th smallest letter(s) of p with i . Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Reduction Given a string of letters p = p 1 ... p n , the reduction of p is the string obtained by replacing the i th smallest letter(s) of p with i . For example, the reduction of 2674425 is Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Reduction Given a string of letters p = p 1 ... p n , the reduction of p is the string obtained by replacing the i th smallest letter(s) of p with i . For example, the reduction of 2674425 is 1 • • •• 1 • . Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Reduction Given a string of letters p = p 1 ... p n , the reduction of p is the string obtained by replacing the i th smallest letter(s) of p with i . For example, the reduction of 2674425 is 1 •• 221 • . Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Reduction Given a string of letters p = p 1 ... p n , the reduction of p is the string obtained by replacing the i th smallest letter(s) of p with i . For example, the reduction of 2674425 is 1 •• 2213 . Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Reduction Given a string of letters p = p 1 ... p n , the reduction of p is the string obtained by replacing the i th smallest letter(s) of p with i . For example, the reduction of 2674425 is 14 • 2213 . Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Reduction Given a string of letters p = p 1 ... p n , the reduction of p is the string obtained by replacing the i th smallest letter(s) of p with i . For example, the reduction of 2674425 is 1452213 . Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Pattern Avoidance in Words Given w ∈ [ k ] n , and p = p 1 . . . p m , w contains p if there is 1 ≤ i 1 < · · · < i m ≤ n so that w i 1 . . . w i m reduces to p . Otherwise w avoids p . Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Pattern Avoidance in Words Given w ∈ [ k ] n , and p = p 1 . . . p m , w contains p if there is 1 ≤ i 1 < · · · < i m ≤ n so that w i 1 . . . w i m reduces to p . Otherwise w avoids p . e.g. 1452213 contains 312 (1452213 ) 1452213 avoids 212. Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Pattern Avoidance in Words Given w ∈ [ k ] n , and p = p 1 . . . p m , w contains p if there is 1 ≤ i 1 < · · · < i m ≤ n so that w i 1 . . . w i m reduces to p . Otherwise w avoids p . e.g. 1452213 contains 312 (1452213 ) 1452213 avoids 212. Want to count A [ a 1 ,..., a k ] ( { Q } ) := � a i | w has a i i ’s , w avoids q for every q ∈ Q } { w ∈ [ k ] Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Outline Introduction/History 1 Pattern Avoidance in Words Previous Work Prefix Schemes for Words 2 Definitions Examples Success Rate Other Schemes for Words 3 Schemes for Monotone Patterns Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Previous Work for Words Results by... Burstein: initial results, generating functions Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Previous Work for Words Results by... Burstein: initial results, generating functions Albert, Aldred, Atkinson, Handley, Holton: results for specific 3-letter patterns Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Previous Work for Words Results by... Burstein: initial results, generating functions Albert, Aldred, Atkinson, Handley, Holton: results for specific 3-letter patterns Brändén, Mansour: automata for enumeration, for specific k Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Previous Work for Words Results by... Burstein: initial results, generating functions Albert, Aldred, Atkinson, Handley, Holton: results for specific 3-letter patterns Brändén, Mansour: automata for enumeration, for specific k Note: most work is for specific patterns, would like a universal technique that works well regardless of pattern or alphabet size Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Prefix Schemes for Words Pattern Avoidance in Words Other Schemes for Words Previous Work Summary Previous Work for Words Results by... Burstein: initial results, generating functions Albert, Aldred, Atkinson, Handley, Holton: results for specific 3-letter patterns Brändén, Mansour: automata for enumeration, for specific k Note: most work is for specific patterns, would like a universal technique that works well regardless of pattern or alphabet size For permutations, one universal technique is Zeilberger and Vatter’s Enumeration Schemes. Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Definitions Prefix Schemes for Words Examples Other Schemes for Words Success Rate Summary Outline Introduction/History 1 Pattern Avoidance in Words Previous Work Prefix Schemes for Words 2 Definitions Examples Success Rate Other Schemes for Words 3 Schemes for Monotone Patterns Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Definitions Prefix Schemes for Words Examples Other Schemes for Words Success Rate Summary Refinement Main Idea: Can’t always directly find a recurrence to count A [ a 1 ,..., a k ] ( { Q } ) Instead, divide and conquer according to pattern formed by first i letters Look for recurrences between these subsets of A [ a 1 ,..., a k ] ( { Q } ) Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Definitions Prefix Schemes for Words Examples Other Schemes for Words Success Rate Summary Notation When Q is understood, � a i | w has prefix p 1 . . . p l } � � A [ a 1 ,..., a k ] p 1 . . . p l := { w ∈ [ k ] Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Definitions Prefix Schemes for Words Examples Other Schemes for Words Success Rate Summary Notation When Q is understood, � a i | w has prefix p 1 . . . p l } � � A [ a 1 ,..., a k ] p 1 . . . p l := { w ∈ [ k ] and, for 1 ≤ i 1 ≤ · · · ≤ i l ≤ k , � p 1 . . . p l � � a i | A [ a 1 ,..., a k ] := { w ∈ [ k ] i 1 . . . i l w has prefix p 1 . . . p l and i 1 , . . . i l are the first l letters of w } Lara Pudwell Schemes for Pattern-Avoiding Words

Introduction/History Definitions Prefix Schemes for Words Examples Other Schemes for Words Success Rate Summary Refinement Example We have, A [ a 1 ,..., a k ] () = A [ a 1 ,..., a k ] ( 1 ) = A [ a 1 ,..., a k ] ( 12 ) ∪ A [ a 1 ,..., a k ] ( 11 ) ∪ A [ a 1 ,..., a k ] ( 21 ) Lara Pudwell Schemes for Pattern-Avoiding Words