Avoiding the Pattern 31542 Patterns Gaussian Polynomials and - PowerPoint PPT Presentation

Avoiding the Pattern 31542 Lara Pudwell Barred Avoiding the Pattern 31542 Patterns Gaussian Polynomials and Partitions The sequence Lara Pudwell A047970 Valparaiso University The pattern 31542 Conclusion Permutation Patterns 2009 July 16, 2009

Bar Notation Avoiding the Pattern 31542 Lara Pudwell A barred permutation pattern is a permutation where each Barred number may or may not have a bar over it. Patterns E.g. p = 31542 is a barred pattern. Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation Avoiding the Pattern 31542 Lara Pudwell A barred permutation pattern is a permutation where each Barred number may or may not have a bar over it. Patterns E.g. p = 31542 is a barred pattern. Gaussian Polynomials and Partitions The sequence A barred pattern p encodes two permutation patterns, A047970 1 The smaller pattern p s formed by the unbarred The pattern 31542 letters of p . Conclusion (in this case, 542 forms a 321 pattern.) 2 The larger pattern p ℓ formed by all letters of p . (in this case, 31542.)

Bar Notation Avoiding the Pattern 31542 Lara Pudwell We say that permutation π avoids the barred pattern p iff every copy of p s in π is part of a copy of p ℓ in π . Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation Avoiding the Pattern 31542 Lara Pudwell We say that permutation π avoids the barred pattern p iff every copy of p s in π is part of a copy of p ℓ in π . Barred Patterns Gaussian Example: p = 31542 Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation Avoiding the Pattern 31542 Lara Pudwell We say that permutation π avoids the barred pattern p iff every copy of p s in π is part of a copy of p ℓ in π . Barred Patterns Gaussian Example: p = 31542 Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation Avoiding the Pattern 31542 Lara Pudwell We say that permutation π avoids the barred pattern p iff every copy of p s in π is part of a copy of p ℓ in π . Barred Patterns Gaussian Example: p = 31542 Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � � � S n ( 132 ) Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � � � S n ( 132 ) Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � � � S n ( 132 ) Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � � � S n ( 132 ) Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion = |{ π ∈ S n | π 1 = 1 }|

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � = B n � � � � � S n ( 132 ) � S n ( 1423 ) Gaussian ( n th Bell number) Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion = |{ π ∈ S n | π 1 = 1 }|

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � = B n � � � � � S n ( 132 ) � S n ( 1423 ) Gaussian ( n th Bell number) Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion = |{ π ∈ S n | π 1 = 1 }|

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � = B n � � � � � S n ( 132 ) � S n ( 1423 ) Gaussian ( n th Bell number) Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion = |{ π ∈ S n | π 1 = 1 }|

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � = B n � � � � � S n ( 132 ) � S n ( 1423 ) Gaussian ( n th Bell number) Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion = |{ π ∈ S n | π 1 = 1 }|

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � = B n � � � � � S n ( 132 ) � S n ( 1423 ) Gaussian ( n th Bell number) Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion = |{ π ∈ S n | π 1 = 1 }|

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � = B n � � � � � S n ( 132 ) � S n ( 1423 ) Gaussian ( n th Bell number) Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion = |{ π ∈ S n | π 1 = 1 }|

Barred Pattern Avoidance Avoiding the Pattern 31542 Lara Pudwell Some nice examples of barred pattern avoidance include: Barred Patterns � = ( n − 1 )! � = B n � � � � � S n ( 132 ) � S n ( 1423 ) Gaussian ( n th Bell number) Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion = |{ π ∈ S n | π 1 = 1 }|

Barred Pattern Sightings Avoiding the Pattern 31542 Lara Pudwell West, 1990: A permutation is 2-stack sortable if and Barred only if it avoids 2341 and 35241. Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Sightings Avoiding the Pattern 31542 Lara Pudwell West, 1990: A permutation is 2-stack sortable if and Barred only if it avoids 2341 and 35241. Patterns Gaussian Bousquet-Melou and Butler, 2006: A permutation is Polynomials and Partitions forest-like if and only if it avoids 1324 and 21354. The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Sightings Avoiding the Pattern 31542 Lara Pudwell West, 1990: A permutation is 2-stack sortable if and Barred only if it avoids 2341 and 35241. Patterns Gaussian Bousquet-Melou and Butler, 2006: A permutation is Polynomials and Partitions forest-like if and only if it avoids 1324 and 21354. The sequence A047970 Bousquet-Melou, Claesson, Dukes, and Kitaev, 2008: The pattern Fixed points of the map between ascent sequences 31542 and modified ascent sequences are in bijection with Conclusion permutations which avoid 31524.

Barred Pattern Sightings Avoiding the Pattern 31542 Lara Pudwell West, 1990: A permutation is 2-stack sortable if and Barred only if it avoids 2341 and 35241. Patterns Gaussian Bousquet-Melou and Butler, 2006: A permutation is Polynomials and Partitions forest-like if and only if it avoids 1324 and 21354. The sequence A047970 Bousquet-Melou, Claesson, Dukes, and Kitaev, 2008: The pattern Fixed points of the map between ascent sequences 31542 and modified ascent sequences are in bijection with Conclusion permutations which avoid 31524. Burstein and Lankham, 2006: A permutation is a reverse patience word if and only if it avoids 3 − 1 − 42.

Observations Avoiding the Pattern 31542 Based on computation: Lara Pudwell Conjecture: If q is a barred pattern of length k with Barred Patterns k − 2 bars then either S n ( q ) = 1 or Gaussian S n ( q ) = ( n − ( k − 2 ))! . Polynomials and Partitions Conjecture: S n ( 31542 ) gives the number of ordered The sequence A047970 factorizations over the Gaussian polynomials. (OEIS The pattern A047970) 31542 Conclusion Conjecture: S n ( 14352 ) has generating function 1 � ( 1 − x ) n ) ( 1 / 2 ) n + 1 (OEIS A122993). n ≥ 0 x ( 1 − There are at least 19 new sequences obtained by counting S n ( q ) , where q is a barred pattern of length 5.

Observations Avoiding the Pattern 31542 Based on computation: Lara Pudwell (Asinowski, PP2008) If q is a barred pattern of length k Barred Patterns with k − 2 bars then either S n ( q ) = 1 or Gaussian S n ( q ) = ( n − ( k − 2 ))! . Polynomials and Partitions Conjecture: S n ( 31542 ) gives the number of ordered The sequence A047970 factorizations over the Gaussian polynomials. (OEIS The pattern A047970) 31542 Conclusion Conjecture: S n ( 14352 ) has generating function 1 � ( 1 − x ) n ) ( 1 / 2 ) n + 1 (OEIS A122993). n ≥ 0 x ( 1 − There are at least 19 new sequences obtained by counting S n ( q ) , where q is a barred pattern of length 5.