Enumeration schemes for permutation patterns dashed permutation - PowerPoint PPT Presentation

Enumeration schemes for dashed Enumeration schemes for permutation patterns dashed permutation patterns Lara Pudwell Dashed Patterns Definition Andrew Baxter Brief History Rutgers University Enumeration Schemes Definition & Notation Lara Pudwell ∗ Example Algorithm Valparaiso University Usefulness Results Guaranteed Schemes Success Rates Sequences Permutation Patterns 2011 June 21, 2011

Dashed Patterns Enumeration schemes for dashed permutation patterns Dashed Pattern Lara Pudwell A dashed permutation pattern is a permutation π ∈ S n where each pair of consecutive numbers may or may not have a dash Dashed Patterns between them. (also called generalized or vincular patterns) Definition Brief History Enumeration Schemes No dash – numbers must appear adjacently Definition & Notation Dash – numbers may appear arbitrarily far apart. Example Algorithm Usefulness Results Example Guaranteed Schemes 251643 contains 1 − 2 − 3 and 12 − 3 but avoids 1 − 23. Success Rates Sequences

Dashed Patterns – Brief History Enumeration schemes for dashed permutation patterns Introduced by Babson and Steingr´ ımsson (2000) in Lara Pudwell systematic search for Mahonian permutation statistics Dashed Selected other work: Patterns Definition Claesson – results for length 3 patterns with two adjacent Brief History letters (2001) Enumeration Schemes Elizalde and Noy – consecutive patterns (2003) Definition & Notation Elizalde – asymptotic enumeration (2006) Example Algorithm Further enumerative results by Bernini, Ferrari, Kitaev, Usefulness Mansour, Pergola, Pinzani Results Guaranteed Survey article by Steingr´ ımsson in PP2007 proceedings Schemes Success Rates Sequences

Enumeration Scheme Definition/Notation Enumeration schemes for dashed Definition (informal) permutation patterns An enumeration scheme is an encoding for a family of Lara Pudwell recurrence relations enumerating members of a family of sets. Dashed Patterns Definition Brief History Notation: Prefix Pattern Enumeration Schemes Definition & S n ( Q )[ p ] = { π ∈ S n ( Q ) | π 1 · · · π | p | ∼ p } Notation Example Algorithm Usefulness Results Guaranteed Note Schemes Success Rates Sequences S n ( Q ) = S n ( Q )[1] = S n ( Q )[12] ∪ S n ( Q )[21] = · · ·

Enumeration Scheme Notation Enumeration schemes for dashed permutation patterns Notation: Children Lara Pudwell The children of prefix p ∈ S n are the length n + 1 permutations Dashed Patterns whose first n letters are order-isomorphic to p . Definition Brief History For example, the children of 213 are { 2134 , 2143 , 3142 , 3241 } . Enumeration Schemes Definition & Note Notation Example Algorithm � Usefulness S n ( Q )[ p ] = S n ( Q )[ c ] Results c ∈ children ( p ) Guaranteed Schemes Success Rates Sequences

Enumeration Scheme Notation Enumeration schemes for dashed permutation patterns Notation: Prefix Pattern with Specified Letters Lara Pudwell Dashed S n ( Q )[ p ; w ] = { π ∈ S n ( Q )[ p ] | π 1 · · · π | p | = w } Patterns Definition Brief History Enumeration Schemes Note Definition & Notation Example n Algorithm � = � � � � � Usefulness � S n ( Q )[1] � S n ( Q )[1; a ] � Results a =1 Guaranteed Schemes Success Rates Sequences

Enumeration Scheme Definitions Enumeration schemes for dashed Definition: Deletion Map d R permutation patterns For a set of indices R , d R ( π ) is the permutation obtained by Lara Pudwell deleting π r for all r ∈ R and reducing. Dashed Patterns For example, d { 1 , 3 } (53241) = red (341) = 231. Definition Brief History Enumeration Definition: Reversibly Deletable Schemes Definition & A set of indices R is reversibly deletable for prefix p with Notation Example Algorithm respect to Q if Usefulness Results Guaranteed d R : S n ( Q )[ p ; w ] → S n −| R | ( Q )[ d R ( p ); d R ( w )] Schemes Success Rates Sequences is a bijection for all w such that S n ( Q )[ p ; w ] � = ∅ .

Enumeration Scheme Example Enumeration Avoiding { 23 − 1 } schemes for dashed permutation patterns Lara Pudwell S n ( { 23 − 1 } ) = S n ( { 23 − 1 } )[1] = S n ( { 23 − 1 } )[12] ∪ S n ( { 23 − 1 } )[21] Dashed Patterns Consider a member of S n ( { 23 − 1 } )[21]... Definition Brief History Enumeration Schemes Definition & Notation Example Algorithm Usefulness Results Guaranteed Schemes Success Rates Sequences

Enumeration Scheme Example Enumeration Avoiding { 23 − 1 } schemes for dashed permutation patterns Lara Pudwell S n ( { 23 − 1 } ) = S n ( { 23 − 1 } )[1] = S n ( { 23 − 1 } )[12] ∪ S n ( { 23 − 1 } )[21] Dashed Patterns Consider a member of S n ( { 23 − 1 } )[21]... Definition Brief History Enumeration Schemes Definition & Notation Example Algorithm Usefulness Results Guaranteed Schemes Success Rates Sequences

Enumeration Scheme Example Enumeration Avoiding { 23 − 1 } schemes for dashed permutation patterns Lara Pudwell S n ( { 23 − 1 } ) = S n ( { 23 − 1 } )[1] = S n ( { 23 − 1 } )[12] ∪ S n ( { 23 − 1 } )[21] Dashed Patterns Consider a member of S n ( { 23 − 1 } )[21]... Definition Brief History Enumeration Schemes Definition & Notation Example Algorithm Usefulness Results Guaranteed Schemes Success Rates Sequences � = � � � � � S n ( { 23 − 1 } )[21; ab ] � S n − 1 ( { 23 − 1 } )[1; b ] So � (i.e. { 1 } is reversibly deletable for p = 21, Q = { 23 − 1 } .)

Two More Definitions... Enumeration schemes for Definition: Spacing Vector dashed permutation Given a word w ∈ [ n ] k , let c i be the i th smallest letter of w , patterns Lara Pudwell c 0 = 0 and c k +1 = n + 1. The vector � g ( n , w ) = � c 1 − c 0 − 1 , c 2 − c 1 − 1 , . . . , c k +1 − c k − 1 � Dashed Patterns is the spacing vector of w . Definition Brief History For example, w = 16372 ∈ [8] 5 has Enumeration Schemes c 0 = 0, c 1 = 1, c 2 = 2, c 3 = 3, c 4 = 6, c 5 = 7, c 6 = 9. Definition & Notation Example Thus, � g (8 , 16372) = � 0 , 0 , 0 , 2 , 0 , 1 � . Algorithm Usefulness Results Definition: Gap Vector Guaranteed Schemes Success Rates v ∈ N k +1 is a gap vector for prefix p with respect to set of � Sequences patterns Q if for all n , S n ( Q )[ p ; w ] = ∅ for all w such that g ( n , w ) ≥ � v componentwise. �

Enumeration Scheme Example Enumeration Avoiding { 23 − 1 } schemes for dashed permutation patterns Lara Pudwell S n ( { 23 − 1 } ) = S n ( { 23 − 1 } )[1] = S n ( { 23 − 1 } )[12] ∪ S n ( { 23 − 1 } )[21] Dashed Patterns Consider a member of S n ( { 23 − 1 } )[12]... Definition Brief History Enumeration Schemes Definition & Notation Example Algorithm Usefulness Results Guaranteed Schemes Success Rates Sequences

Enumeration Scheme Example Enumeration Avoiding { 23 − 1 } schemes for dashed permutation patterns Lara Pudwell S n ( { 23 − 1 } ) = S n ( { 23 − 1 } )[1] = S n ( { 23 − 1 } )[12] ∪ S n ( { 23 − 1 } )[21] Dashed Patterns Consider a member of S n ( { 23 − 1 } )[12]... Definition Brief History Enumeration Schemes Definition & Notation ( � 1 , 0 , 0 � is a gap vector Example Algorithm for p = 12, Q = { 23 − 1 } ) Usefulness Results Guaranteed Schemes Success Rates Sequences

Enumeration Scheme Example Enumeration Avoiding { 23 − 1 } schemes for dashed permutation patterns Lara Pudwell S n ( { 23 − 1 } ) = S n ( { 23 − 1 } )[1] = S n ( { 23 − 1 } )[12] ∪ S n ( { 23 − 1 } )[21] Dashed Patterns Consider a member of S n ( { 23 − 1 } )[12]... Definition Brief History Enumeration Schemes Definition & Notation ( � 1 , 0 , 0 � is a gap vector Example Algorithm for p = 12, Q = { 23 − 1 } ) Usefulness Results Guaranteed Schemes Success Rates Sequences � 0 a > 1 � = � � So � S n ( { 23 − 1 } )[12; ab ]

Enumeration Scheme Example Enumeration Avoiding { 23 − 1 } schemes for dashed permutation patterns Lara Pudwell S n ( { 23 − 1 } ) = S n ( { 23 − 1 } )[1] = S n ( { 23 − 1 } )[12] ∪ S n ( { 23 − 1 } )[21] Dashed Patterns Consider a member of S n ( { 23 − 1 } )[12]... Definition Brief History Enumeration Schemes Definition & Notation ( � 1 , 0 , 0 � is a gap vector Example Algorithm for p = 12, Q = { 23 − 1 } ) Usefulness Results Guaranteed Schemes Success Rates Sequences � 0 a > 1 � = � � So � S n ( { 23 − 1 } )[12; ab ]

Enumeration Scheme Example Enumeration Avoiding { 23 − 1 } schemes for dashed permutation patterns Lara Pudwell S n ( { 23 − 1 } ) = S n ( { 23 − 1 } )[1] = S n ( { 23 − 1 } )[12] ∪ S n ( { 23 − 1 } )[21] Dashed Patterns Consider a member of S n ( { 23 − 1 } )[12]... Definition Brief History Enumeration Schemes Definition & Notation ( � 1 , 0 , 0 � is a gap vector Example Algorithm for p = 12, Q = { 23 − 1 } ) Usefulness Results Guaranteed Schemes Success Rates Sequences � 0 a > 1 � = � � So � S n ( { 23 − 1 } )[12; ab ]