Pattern-Avoiding Permutations Enumeration Schemes Summary An Introduction to Enumeration Schemes Lara Pudwell Valparaiso University Lara.Pudwell@valpo.edu AWM Workshop Washington, DC January 8, 2009
Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Reduction Given a string of numbers p = p 1 · · · p n , the reduction of p is the string obtained by replacing the i th smallest number of p with i . For example, the reduction of 26745 is
Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Reduction Given a string of numbers p = p 1 · · · p n , the reduction of p is the string obtained by replacing the i th smallest number of p with i . For example, the reduction of 26745 is 1 •••• .
Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Reduction Given a string of numbers p = p 1 · · · p n , the reduction of p is the string obtained by replacing the i th smallest number of p with i . For example, the reduction of 26745 is 1 •• 2 • .
Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Reduction Given a string of numbers p = p 1 · · · p n , the reduction of p is the string obtained by replacing the i th smallest number of p with i . For example, the reduction of 26745 is 1 •• 23 .
Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Reduction Given a string of numbers p = p 1 · · · p n , the reduction of p is the string obtained by replacing the i th smallest number of p with i . For example, the reduction of 26745 is 14 • 23 .
Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Reduction Given a string of numbers p = p 1 · · · p n , the reduction of p is the string obtained by replacing the i th smallest number of p with i . For example, the reduction of 26745 is 14523 .
Pattern-Avoiding Permutations Enumeration Schemes Summary Definitions Pattern Avoidance in Permutations Given p ∈ S n and q ∈ S m , we say p contains q if there are 1 ≤ i 1 < · · · < i m ≤ n such that p i 1 · · · p i m reduces to q . Otherwise, p avoids q . p = 21354 contains 132. (since 21354 reduces to 132.) p = 21354 avoids 321. (since p has no decreasing subsequence of length 3.)
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Two Questions Easy: Given p ∈ S n , what patterns does p contain? Hard: Given q ∈ S m , Let S n ( q ) = { p ∈ S n | p avoids q } . Find an expression for | S n ( q ) | .
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Patterns of length 1, 2, and 3 � 1 n = 0 | S n ( 1 ) | = 0 n > 0
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Patterns of length 1, 2, and 3 � 1 n = 0 | S n ( 1 ) | = | S n ( 12 ) | = | S n ( 21 ) | = 1 , n ≥ 0 0 n > 0
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Patterns of length 1, 2, and 3 � 1 n = 0 | S n ( 1 ) | = | S n ( 12 ) | = | S n ( 21 ) | = 1 , n ≥ 0 0 n > 0 Graph of p ∈ S n ( 132 )
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Patterns of length 1, 2, and 3 � 1 n = 0 | S n ( 1 ) | = | S n ( 12 ) | = | S n ( 21 ) | = 1 , n ≥ 0 0 n > 0 Graph of p ∈ S n ( 132 )
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Patterns of length 1, 2, and 3 � 1 n = 0 | S n ( 1 ) | = | S n ( 12 ) | = | S n ( 21 ) | = 1 , n ≥ 0 0 n > 0 Graph of p ∈ S n ( 132 ) So, | S n ( 132 ) | = � n i = 1 | S i − 1 ( 132 ) | · | S n − i ( 132 ) |
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Patterns of length 1, 2, and 3 � 1 n = 0 | S n ( 1 ) | = | S n ( 12 ) | = | S n ( 21 ) | = 1 , n ≥ 0 0 n > 0 Graph of p ∈ S n ( 132 ) � 2 n � So, | S n ( 132 ) | = � n n i = 1 | S i − 1 ( 132 ) | · | S n − i ( 132 ) | = n + 1 = C n In fact, | S n ( q ) | = C n if q is any permutation of length 3.
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Patterns of Length 4 There are 24 patterns of length 4. Using several clever bijections, we can narrow our work to 3 cases: S n ( 1342 ) , S n ( 1234 ) , and S n ( 1324 ) .
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results Patterns of Length 4 There are 24 patterns of length 4. Using several clever bijections, we can narrow our work to 3 cases: S n ( 1342 ) , S n ( 1234 ) , and S n ( 1324 ) . 1 2 3 4 5 6 7 8 ∼ 8 n | S n ( 1342 ) | 1 2 6 23 103 512 2740 15485 ∼ 9 n | S n ( 1234 ) | 1 2 6 23 103 513 2761 15767 ∼ 9 . 3 n | S n ( 1324 ) | 1 2 6 23 103 513 2762 15793
Pattern-Avoiding Permutations Enumeration Schemes Summary Counting Results For Permutations Most techniques studying | S n ( q ) | finds formulas for a specific q . 1998: Zeilberger’s prefix enumeration schemes , i.e. a system of recurrences to count | S n ( q ) | . 2005: Vatter’s modified schemes automate the enumeration of | S n ( q ) | for even more patterns q .
Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Refinement Notation Goal: Divide S n ( q ) into subsets. � � � π avoids q � � � S n q ; p 1 · · · p l := π ∈ S n � π has prefix p 1 · · · p l and π avoids q � � � q ; p 1 · · · p l � S n := π ∈ S n π has prefix p 1 · · · p l � i 1 · · · i l � π = i 1 · · · i l π l + 1 · · · π n � If p 1 · · · p l − 1 reduces to p ∗ 1 · · · p ∗ l − 1 , then p = p 1 · · · p l is called a refinement if p ∗ . e.g p = 2413 is a refinement of p ∗ = 231.
� � Pattern-Avoiding Permutations Enumeration Schemes Summary Divide Refinement Example For any pattern q , S n ( q ) = S n ( q ; 1 ) = S n ( q ; 12 ) ∪ S n ( q ; 21 ) . or graphically: ∅ 1 � � � � � � � � � � � � � � � � � � � 12 21
Pattern-Avoiding Permutations Enumeration Schemes Summary Conquer Reversibly Deletable Positions Reversibly Deletable Positions Given a pattern q and a prefix p , position r is reversibly deletable if Deleting p r from π ∈ S n ( q ; p 1 · · · p l ) produces a q -avoiding permutation of length n − 1, and Inserting p r into π ∈ S n − 1 ( q ; p 1 · · · p r − 1 p r + 1 · · · p l ) produces a q avoiding permutation of length n . In other words, deleting and re-inserting p r gives a bijection between S n ( q ; p 1 · · · p l ) and S n − 1 ( q ; p 1 · · · p r − 1 p r + 1 · · · p l ) , and | S n ( q ; p 1 · · · p l ) | = | S n − 1 ( q ; p 1 · · · p r − 1 p r + 1 · · · p l ) | .
� � Pattern-Avoiding Permutations Enumeration Schemes Summary Conquer Reversibly Deletable Example Graphically: ∅ 1 � � � � � � � � � � � � � � � � � � � 12 21
� � � Pattern-Avoiding Permutations Enumeration Schemes Summary Conquer Reversibly Deletable Example Graphically: ∅ 1 � d 1 � � � � � � � � � � � � � � � � � � 12 21
� � � Pattern-Avoiding Permutations Enumeration Schemes Summary Conquer Gap Vectors Gap vectors give a condition for which choices of i 1 , . . . , i l yield � � �� q ; p 1 · · · p r · · · p l � � � S n � = 0 . � � i 1 · · · i r · · · i l Graphically: ∅ 1 � d 1 � � � � � � � � � � � � � � � � � � 12 21
� � � Pattern-Avoiding Permutations Enumeration Schemes Summary Conquer Gap Vectors Gap vectors give a condition for which choices of i 1 , . . . , i l yield � � �� q ; p 1 · · · p r · · · p l � � � S n � = 0 . � � i 1 · · · i r · · · i l Graphically: ∅ 1 � d 1 � � � � � � � � � � � � � � � � � � 12 21 ≥ ( 0 , 1 , 0 )
� � � � Pattern-Avoiding Permutations Enumeration Schemes Summary Conquer Gap Vectors Gap vectors give a condition for which choices of i 1 , . . . , i l yield � � �� q ; p 1 · · · p r · · · p l � � � S n � = 0 . � � i 1 · · · i r · · · i l Graphically: ∅ 1 d 2 � d 1 � � � � � � � � � � � � � � � � � � 12 21 ≥ ( 0 , 1 , 0 )
Pattern-Avoiding Permutations Enumeration Schemes Summary Enumeration Schemes Enumeration Scheme Definition An enumeration scheme is a set of triples [ p i , R i , G i ] such that for each triple p i is a reduced prefix of length n R i a subset of { 1 , . . . , n } G i is a set of vectors of length n + 1 and either R i is non-empty or all refinements of p i are also in the scheme.
Pattern-Avoiding Permutations Enumeration Schemes Summary Enumeration Schemes Enumeration Scheme Definition An enumeration scheme is a set of triples [ p i , R i , G i ] such that for each triple p i is a reduced prefix of length n (prefix) R i a subset of { 1 , . . . , n } (reversibly deletable positions) G i is a set of vectors of length n + 1 (gap vectors) and either R i is non-empty or all refinements of p i are also in the scheme.
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