Pattern avoidance in double lists Lara Pudwell Introduction Length - PowerPoint PPT Presentation

Pattern avoidance in double lists Pattern avoidance in double lists Lara Pudwell Introduction Length 4 Patterns 1342 2143 Lara Pudwell 1423, 1432, 1243 1234 faculty.valpo.edu/lpudwell 2413 1324 Summary joint work with Charles Cratty (Westminster College) Samuel Erickson (Minnesota State Moorhead) Frehiwet Negassi (St. Joseph’s College) AMS Central Fall Sectional Meeting University of Wisconsin – Eau Claire September 21, 2014

Pattern avoidance Variations on Pattern Avoidance in double lists Lara Pudwell Introduction Length 4 Patterns 1342 ◮ in words (Burstein 1998, and others) 2143 1423, 1432, 1243 ◮ in centrosymmetric permutations (Egge 2007; Barnabei, 1234 2413 Bonetti, Silimbani 2010) 1324 Summary ◮ in centrosymmetric words (Ferrari 2011) ◮ in circular permutations (Callan 2002, Vella 2003) ◮ Here, for example, 1324 = 3241 = 2413 = 4132. Our study: pattern avoidance in words with a special kind of symmetry/repetition.

Pattern avoidance Definitions/Notation in double lists Lara Pudwell ◮ S n is the set of permutations of length n . Introduction Length 4 Patterns S 3 = { 123 , 132 , 213 , 231 , 312 , 321 } . 1342 2143 1423, 1432, 1243 1234 2413 ◮ D n = { ππ | π ∈ S n } . 1324 Summary D 3 = { 123123 , 132132 , 213213 , 231231 , 312312 , 321321 } . ◮ D n ( ρ ) = { σ | σ ∈ D n and σ avoids ρ } . D 3 ( 12 ) = ∅ . D 3 ( 123 ) = { 321321 } . Goal: Characterize D n ( ρ ) /compute |D n ( ρ ) | where ρ ∈ S 4 .

Pattern avoidance Warmup in double lists Lara Pudwell Introduction ◮ D n ( 1 ) = ∅ for n ≥ 1. Length 4 Patterns 1342 ◮ D n ( 12 ) = D n ( 21 ) = ∅ for n ≥ 2. 2143 1423, 1432, 1243 1234 2413 1324 Summary

Pattern avoidance Warmup in double lists Lara Pudwell Introduction ◮ D n ( 1 ) = ∅ for n ≥ 1. Length 4 Patterns 1342 ◮ D n ( 12 ) = D n ( 21 ) = ∅ for n ≥ 2. 2143 1423, 1432, 1243 1234 2413 ◮ |D n ( ρ ) | = |D n ( ρ r ) | = |D n ( ρ c ) | 1324 � in general. � D n ( ρ − 1 ) � � Summary but |D n ( ρ ) | � =

Pattern avoidance Warmup in double lists Lara Pudwell Introduction ◮ D n ( 1 ) = ∅ for n ≥ 1. Length 4 Patterns 1342 ◮ D n ( 12 ) = D n ( 21 ) = ∅ for n ≥ 2. 2143 1423, 1432, 1243 1234 2413 ◮ |D n ( ρ ) | = |D n ( ρ r ) | = |D n ( ρ c ) | 1324 � in general. � D n ( ρ − 1 ) � � Summary but |D n ( ρ ) | � = ◮ D n ( 123 ) = { n · · · 1 n · · · 1 } for n ≥ 3.  { 11 } n = 1     { 1212 , 2121 } n = 2   ◮ D n ( 132 ) = . { 231231 } n = 3     ∅ n ≥ 4  

Pattern avoidance Length 4 Trivial Wilf Classes in double lists Lara Pudwell {|D n ( ρ ) |} 10 Pattern ρ Introduction n = 1 Length 4 Patterns 1342, 2431, 1, 2, 6, 12, 15, 15, 15, 15, 15, 15 1342 3124, 4213 2143 1423, 1432, 1243 2143, 3412 1, 2, 6, 12, 13, 14, 16, 18, 20, 22 1234 2413 1423, 2314, 1324 1, 2, 6, 12, 17, 23, 27, 30, 33, 36 3241, 4132 Summary 1432, 2341, 1, 2, 6, 12, 17, 23, 31, 40, 50, 61 3214, 4123 1243, 2134, 1, 2, 6, 12, 19, 25, 34, 44, 55, 67 3421, 4312 2413, 3142 1, 2, 6, 12, 18, 29, 47, 76, 123, 199 1324, 4231 1, 2, 6, 12, 21, 38, 69, 126, 232, 427 1234, 4321 1, 2, 6, 12, 27, 58, 121, 248, 503, 1014 Contrast: For large n , |S n ( 1342 ) | < |S n ( 1234 ) | < |S n ( 1324 ) | .

D n ( 1342 ) Pattern avoidance in double lists (1, 2, 6, 12, 15, 15, 15, 15, 15, 15,. . . ) Lara Pudwell Introduction Length 4 Patterns 1342 2143 1423, 1432, 1243 1234 2413 1324 Summary

D n ( 1342 ) Pattern avoidance in double lists (1, 2, 6, 12, 15, 15, 15, 15, 15, 15,. . . ) Lara Pudwell Introduction Length 4 Patterns 1342 2143 1423, 1432, 1243 1234 2413 1324 Summary

D n ( 2143 ) Pattern avoidance in double lists (1, 2, 6, 12, 13, 14, 16, 18, 20, 22,. . . ) Lara Pudwell Case 1: ( . . . , n , 1 , . . . , n , 1 , . . . ) ( n + 1 lists) Introduction Length 4 Patterns 1342 2143 1423, 1432, 1243 1234 2413 1324 Case 2: ( . . . , n , a , 1 , . . . , n , a , 1 , . . . ) ( n − 2 lists) Summary Case 3: ( . . . , 1 , . . . , n , . . . , 1 , . . . , n , . . . ) (3 lists) ( n + 1 ) + ( n − 2 ) + 3 = 2 n + 2 .

D n ( ρ ) , ρ ∈ { 1423 , 1432 , 1243 } Pattern avoidance in double lists (More case analysis...) Lara Pudwell For ρ = 1423, (1, 2, 6, 12, 17, 23, 27, 30, 33, 36,. . . ) Introduction Length 4 Patterns 1342 2143 1423, 1432, 1243 1234 2413 For ρ = 1432, (1, 2, 6, 12, 17, 23, 31, 40, 50, 61,. . . ) 1324 Summary For ρ = 1243, (1, 2, 6, 12, 19, 25, 34, 44, 55, 67,. . . )

D n ( ρ ) , ρ ∈ { 1423 , 1432 , 1243 } Pattern avoidance in double lists (More case analysis...) Lara Pudwell For ρ = 1423, (1, 2, 6, 12, 17, 23, 27, 30, 33, 36,. . . ) Introduction |D n ( 1423 ) | = 3 n + 6 ( n ≥ 7) Length 4 Patterns 1342 2143 1423, 1432, 1243 1234 2413 For ρ = 1432, (1, 2, 6, 12, 17, 23, 31, 40, 50, 61,. . . ) 1324 Summary For ρ = 1243, (1, 2, 6, 12, 19, 25, 34, 44, 55, 67,. . . )

D n ( ρ ) , ρ ∈ { 1423 , 1432 , 1243 } Pattern avoidance in double lists (More case analysis...) Lara Pudwell For ρ = 1423, (1, 2, 6, 12, 17, 23, 27, 30, 33, 36,. . . ) Introduction |D n ( 1423 ) | = 3 n + 6 ( n ≥ 7) Length 4 Patterns 1342 2143 1423, 1432, 1243 1234 2413 For ρ = 1432, (1, 2, 6, 12, 17, 23, 31, 40, 50, 61,. . . ) 1324 Summary |D n ( 1432 ) | = |D n − 1 ( 1432 ) | + ( n + 1 ) ( n ≥ 7) 2 n 2 + 3 |D n ( 1432 ) | = 1 2 n − 4 ( n ≥ 6) For ρ = 1243, (1, 2, 6, 12, 19, 25, 34, 44, 55, 67,. . . )

D n ( ρ ) , ρ ∈ { 1423 , 1432 , 1243 } Pattern avoidance in double lists (More case analysis...) Lara Pudwell For ρ = 1423, (1, 2, 6, 12, 17, 23, 27, 30, 33, 36,. . . ) Introduction |D n ( 1423 ) | = 3 n + 6 ( n ≥ 7) Length 4 Patterns 1342 2143 1423, 1432, 1243 1234 2413 For ρ = 1432, (1, 2, 6, 12, 17, 23, 31, 40, 50, 61,. . . ) 1324 Summary |D n ( 1432 ) | = |D n − 1 ( 1432 ) | + ( n + 1 ) ( n ≥ 7) 2 n 2 + 3 |D n ( 1432 ) | = 1 2 n − 4 ( n ≥ 6) For ρ = 1243, (1, 2, 6, 12, 19, 25, 34, 44, 55, 67,. . . ) |D n ( 1243 ) | = |D n − 1 ( 1243 ) | + ( n + 2 ) ( n ≥ 7) 2 n 2 + 5 |D n ( 1243 ) | = 1 2 n − 8 ( n ≥ 6)

D n ( 1234 ) Pattern avoidance in double lists (1, 2, 6, 12, 27, 58, 121, 248, 503, 1014,. . . ) Lara Pudwell For n ≥ 4, in OEIS, “Number of different permutations of a Introduction deck of n cards that can be produced by a single shuffle”. Length 4 Patterns 1342 1. Begin with ordered deck n · · · 1. 2143 1423, 1432, 1243 2. Cut. 1234 2413 3. Each card either comes from upper or lower partial deck. 1324 Summary

D n ( 1234 ) Pattern avoidance in double lists (1, 2, 6, 12, 27, 58, 121, 248, 503, 1014,. . . ) Lara Pudwell For n ≥ 4, in OEIS, “Number of different permutations of a Introduction deck of n cards that can be produced by a single shuffle”. Length 4 Patterns 1342 1. Begin with ordered deck n · · · 1. 2143 1423, 1432, 1243 2. Cut. 1234 2413 3. Each card either comes from upper or lower partial deck. 1324 Summary Picture of 1234-avoiding double list:

D n ( 1234 ) Pattern avoidance in double lists (1, 2, 6, 12, 27, 58, 121, 248, 503, 1014,. . . ) Lara Pudwell For n ≥ 4, in OEIS, “Number of different permutations of a Introduction deck of n cards that can be produced by a single shuffle”. Length 4 Patterns 1342 1. Begin with ordered deck n · · · 1. 2143 1423, 1432, 1243 2. Cut. 1234 2413 3. Each card either comes from upper or lower partial deck. 1324 Summary Picture of 1234-avoiding double list: There are 2 n strings on { U , L } n , but the ( n + 1 ) decks of the form U · · · UL · · · L are all equivalent to the original deck. |D n ( 1234 ) | = 2 n − n for n ≥ 4 .

D n ( 2413 ) Pattern avoidance in double lists (1, 2, 6, 12, 18, 29, 47, 76, 123, 199,. . . ) Lara Pudwell Key observation: Introduction 1 must appear in position 1, n − 2, n − 1, or n . Length 4 Patterns ◮ e.g. If 1 is in position n − 2: 1342 2143 1423, 1432, 1243 must begin with n − 1 or ( n − 2 )( n − 1 ) . 1234 2413 1324 Summary

D n ( 2413 ) (1 in position n − 2 ) Pattern avoidance in double lists In D 5 ( 2413 ) ... Lara Pudwell Introduction Length 4 Patterns 1342 2143 In D 6 ( 2413 ) ... 1423, 1432, 1243 1234 2413 1324 Summary In D 7 ( 2413 ) ...

D n ( 2413 ) Pattern avoidance in double lists (1, 2, 6, 12, 18, 29, 47, 76, 123, 199,. . . ) Lara Pudwell Key observation: 1 must appear in position 1, n − 2, n − 1, Introduction or n . Length 4 Patterns ◮ e.g. If 1 is in position n − 2: must begin with n − 1 or 1342 2143 1423, 1432, 1243 ( n − 2 )( n − 1 ) . 1234 2413 ◮ similar recursions for other positions of 1. 1324 Summary

D n ( 2413 ) Pattern avoidance in double lists (1, 2, 6, 12, 18, 29, 47, 76, 123, 199,. . . ) Lara Pudwell Key observation: 1 must appear in position 1, n − 2, n − 1, Introduction or n . Length 4 Patterns ◮ e.g. If 1 is in position n − 2: must begin with n − 1 or 1342 2143 1423, 1432, 1243 ( n − 2 )( n − 1 ) . 1234 2413 ◮ similar recursions for other positions of 1. 1324 Summary For n ≥ 7, |D n ( 2413 ) | = |D n − 1 ( 2413 ) | + |D n − 2 ( 2413 ) | . i.e. |D n ( 2413 ) | = L n + 2 ( n ≥ 5).

D n ( 1324 ) Pattern avoidance in double lists Lara Pudwell (1, 2, 6, 12, 21, 38, 69, 126, 232, 427,. . . ) Introduction Length 4 Patterns 1342 2143 1423, 1432, 1243 1234 2413 1324 Summary