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

▶
avoiding the pattern 31542
SMART_READER_LITE
LIVE PREVIEW

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

Apr 08, 2024 31 likes •786 views

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

slide-1
SLIDE 1

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Avoiding the Pattern 31542

Lara Pudwell Valparaiso University Permutation Patterns 2009 July 16, 2009

slide-2
SLIDE 2

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation A barred permutation pattern is a permutation where each number may or may not have a bar over it. E.g. p = 31542 is a barred pattern.

slide-3
SLIDE 3

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation A barred permutation pattern is a permutation where each number may or may not have a bar over it. E.g. p = 31542 is a barred pattern. A barred pattern p encodes two permutation patterns,

1 The smaller pattern ps formed by the unbarred

letters of p. (in this case, 542 forms a 321 pattern.)

2 The larger pattern pℓ formed by all letters of p.

(in this case, 31542.)

slide-4
SLIDE 4

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation We say that permutation π avoids the barred pattern p iff every copy of ps in π is part of a copy of pℓ in π.

slide-5
SLIDE 5

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation We say that permutation π avoids the barred pattern p iff every copy of ps in π is part of a copy of pℓ in π. Example: p = 31542

slide-6
SLIDE 6

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation We say that permutation π avoids the barred pattern p iff every copy of ps in π is part of a copy of pℓ in π. Example: p = 31542

slide-7
SLIDE 7

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Bar Notation We say that permutation π avoids the barred pattern p iff every copy of ps in π is part of a copy of pℓ in π. Example: p = 31542

slide-8
SLIDE 8

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!
slide-9
SLIDE 9

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!
slide-10
SLIDE 10

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!
slide-11
SLIDE 11

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!

= |{π ∈ Sn|π1 = 1}|

slide-12
SLIDE 12

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!

= |{π ∈ Sn|π1 = 1}|

  • Sn(1423)
  • = Bn

(nth Bell number)

slide-13
SLIDE 13

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!

= |{π ∈ Sn|π1 = 1}|

  • Sn(1423)
  • = Bn

(nth Bell number)

slide-14
SLIDE 14

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!

= |{π ∈ Sn|π1 = 1}|

  • Sn(1423)
  • = Bn

(nth Bell number)

slide-15
SLIDE 15

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!

= |{π ∈ Sn|π1 = 1}|

  • Sn(1423)
  • = Bn

(nth Bell number)

slide-16
SLIDE 16

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!

= |{π ∈ Sn|π1 = 1}|

  • Sn(1423)
  • = Bn

(nth Bell number)

slide-17
SLIDE 17

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!

= |{π ∈ Sn|π1 = 1}|

  • Sn(1423)
  • = Bn

(nth Bell number)

slide-18
SLIDE 18

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Avoidance Some nice examples of barred pattern avoidance include:

  • Sn(132)
  • = (n − 1)!

= |{π ∈ Sn|π1 = 1}|

  • Sn(1423)
  • = Bn

(nth Bell number)

slide-19
SLIDE 19

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Sightings West, 1990: A permutation is 2-stack sortable if and

  • nly if it avoids 2341 and 35241.
slide-20
SLIDE 20

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Sightings West, 1990: A permutation is 2-stack sortable if and

  • nly if it avoids 2341 and 35241.

Bousquet-Melou and Butler, 2006: A permutation is forest-like if and only if it avoids 1324 and 21354.

slide-21
SLIDE 21

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Sightings West, 1990: A permutation is 2-stack sortable if and

  • nly if it avoids 2341 and 35241.

Bousquet-Melou and Butler, 2006: A permutation is forest-like if and only if it avoids 1324 and 21354. Bousquet-Melou, Claesson, Dukes, and Kitaev, 2008: Fixed points of the map between ascent sequences and modified ascent sequences are in bijection with permutations which avoid 31524.

slide-22
SLIDE 22

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Barred Pattern Sightings West, 1990: A permutation is 2-stack sortable if and

  • nly if it avoids 2341 and 35241.

Bousquet-Melou and Butler, 2006: A permutation is forest-like if and only if it avoids 1324 and 21354. Bousquet-Melou, Claesson, Dukes, and Kitaev, 2008: Fixed points of the map between ascent sequences and modified ascent sequences are in bijection with permutations which avoid 31524. Burstein and Lankham, 2006: A permutation is a reverse patience word if and only if it avoids 3−1−42.

slide-23
SLIDE 23

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Observations Based on computation: Conjecture: If q is a barred pattern of length k with k − 2 bars then either Sn(q) = 1 or Sn(q) = (n − (k − 2))!. Conjecture: Sn(31542) gives the number of ordered factorizations over the Gaussian polynomials. (OEIS A047970) Conjecture: Sn(14352) has generating function

  • n≥0

1 (1−

x (1−x)n )(1/2)n+1 (OEIS A122993).

There are at least 19 new sequences obtained by counting Sn(q), where q is a barred pattern of length 5.

slide-24
SLIDE 24

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Observations Based on computation: (Asinowski, PP2008) If q is a barred pattern of length k with k − 2 bars then either Sn(q) = 1 or Sn(q) = (n − (k − 2))!. Conjecture: Sn(31542) gives the number of ordered factorizations over the Gaussian polynomials. (OEIS A047970) Conjecture: Sn(14352) has generating function

  • n≥0

1 (1−

x (1−x)n )(1/2)n+1 (OEIS A122993).

There are at least 19 new sequences obtained by counting Sn(q), where q is a barred pattern of length 5.

slide-25
SLIDE 25

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Observations Based on computation: (Asinowski, PP2008) If q is a barred pattern of length k with k − 2 bars then either Sn(q) = 1 or Sn(q) = (n − (k − 2))!. Conjecture: Sn(31542) gives the number of ordered factorizations over the Gaussian polynomials. (OEIS A047970) Conjecture: Sn(14352) has generating function

  • n≥0

1 (1−

x (1−x)n )(1/2)n+1 (OEIS A122993).

There are at least 19 new sequences obtained by counting Sn(q), where q is a barred pattern of length 5.

slide-26
SLIDE 26

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

A Little Help from OEIS

slide-27
SLIDE 27

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials Recall that the q-analogue of an integer n is the polynomial [n]q = 1 + q + q2 + · · · + qn−1. From this, we can define the q-factorial as [n]q! = [n]q · [n − 1]q · · · [1]q. and the q-binomial as m n

  • q

= [m]q! [n]q![m − n]q!. These q-binomial coefficients turn out to be polynomials in the variable q, and are called Gaussian Polynomials.

slide-28
SLIDE 28

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials and Partitions The coefficient of qk in m n

  • q

gives the number of partitions of k that fit inside an n × (m − n) box. For example, 4 2

  • q

= 1 + q + 2q2 + q3 + q4.

slide-29
SLIDE 29

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials and Partitions The coefficient of qk in m n

  • q

gives the number of partitions of k that fit inside an n × (m − n) box. For example, 4 2

  • q

= 1 + q + 2q2 + q3 + q4.

slide-30
SLIDE 30

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials and Partitions The coefficient of qk in m n

  • q

gives the number of partitions of k that fit inside an n × (m − n) box. For example, 4 2

  • q

= 1 + q + 2q2 + q3 + q4.

slide-31
SLIDE 31

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials and Partitions The coefficient of qk in m n

  • q

gives the number of partitions of k that fit inside an n × (m − n) box. For example, 4 2

  • q

= 1 + q + 2q2 + q3 + q4.

slide-32
SLIDE 32

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials and Partitions The coefficient of qk in m n

  • q

gives the number of partitions of k that fit inside an n × (m − n) box. For example, 4 2

  • q

= 1 + q + 2q2 + q3 + q4.

slide-33
SLIDE 33

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials and Partitions The coefficient of qk in m n

  • q

gives the number of partitions of k that fit inside an n × (m − n) box. For example, 4 2

  • q

= 1 + q + 2q2 + q3 + q4.

slide-34
SLIDE 34

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials and Partitions The coefficient of qk in m n

  • q

gives the number of partitions of k that fit inside an n × (m − n) box. But what if we wanted to use the full dimensions of the box?

slide-35
SLIDE 35

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Gaussian Polynomials and Partitions By adding a new bottom row and new leftmost column to each partition, the coefficient of qk in m n

  • q

gives the number of partitions of k + m + 1 into m − n + 1 parts with largest part n + 1. We will call these modified partitions.

slide-36
SLIDE 36

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

"Ordered Factorizations over the Gaussian Polynomials" To generate A047970, Arnold considered the modified partitions corresponding to the Gaussian polynomials in the nth row of Pascal’s triangle: n − 1

  • q

, n − 1 1

  • q

, · · · , n − 1 n − 1

  • q

. There are 2n−1 such partitions. How many compositions can be obtained from these partitions?

slide-37
SLIDE 37

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Arnold’s construction for n = 2 Polynomials Partitions Compositions 1

  • q

= 1 1 1

  • q

= 1

slide-38
SLIDE 38

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Arnold’s construction for n = 2 Polynomials Partitions Compositions 1

  • q

= 1 1 1

  • q

= 1

slide-39
SLIDE 39

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Arnold’s construction for n = 2 Polynomials Partitions Compositions 1

  • q

= 1 1 1

  • q

= 1 2 2

slide-40
SLIDE 40

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Arnold’s construction for n = 3 Polynomials Partitions Compositions 2

  • q

= 1 2 1

  • q

= 1 + q 2 2

  • q

= 1 4 5

slide-41
SLIDE 41

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Arnold’s construction for n = 4

Polynomials Partitions Compositions 3

  • q

= 1 3 1

  • q

= 1 + q + q2

  • 3

2

  • q

= 1 + q + q2 3 3

  • q

= 1 8 14

slide-42
SLIDE 42

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

A formula for A047970 There is an easy bijection from the compositions corresponding to m n

  • q

in Arnold’s construction and set partitions of {1, . . . , m − n + 1} into n + 1 ordered blocks.

slide-43
SLIDE 43

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

A formula for A047970 There is an easy bijection from the compositions corresponding to m n

  • q

in Arnold’s construction and set partitions of {1, . . . , m − n + 1} into n + 1 ordered blocks.

slide-44
SLIDE 44

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

A formula for A047970 There is an easy bijection from the compositions corresponding to m n

  • q

in Arnold’s construction and set partitions of {1, . . . , m − n + 1} into n + 1 ordered blocks. This composition corresponds to the set partition {1, 3}, {4}, {}, {2}, {}.

slide-45
SLIDE 45

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

A formula for A047970 The compositions corresponding to the Gaussian polynomial m n

  • q

are in bijection with the ways to partition m − n + 1 elements into n + 1 ordered blocks, where the first block must be non-empty. There are (n + 1)m−n+1 − nm−n+1 such set partitions.

slide-46
SLIDE 46

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

A formula for A047970 Now if we consider m i

  • q

for 0 ≤ i ≤ m, there are

m

  • i=0

(i + 1)m−i+1 − im−i+1 compositions corresponding to the mth row of Pascal’s triangle. Or, with suitable change of variables

n

  • k=0

(n − k + 1)k − (n − k)k.

slide-47
SLIDE 47

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Back to Permutation Patterns Recall, we are interested in Sn(31542).

slide-48
SLIDE 48

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Back to Permutation Patterns Recall, we are interested in Sn(31542). Using brute force computation, for n ≤ 15 we have:

  • Sn(31542)
  • =

n

  • k=0

(n − k + 1)k − (n − k)k. This seems to match Arnold’s sequence. Does the pattern continue?

slide-49
SLIDE 49

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Sn(31542) Observation 1

slide-50
SLIDE 50

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Sn(31542) Observation 1

slide-51
SLIDE 51

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Sn(31542) Observation 1

  • All numbers before the 1 must be in increasing order.
slide-52
SLIDE 52

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Sn(31542) Observation 2

  • All numbers before the 1 must be in increasing order.
slide-53
SLIDE 53

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Sn(31542) Observation 2

  • All numbers before the 1 must be in increasing order.
  • All numbers in the lower right quadrant must be in

increasing order.

slide-54
SLIDE 54

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Subset Definitions This structure allows us to define several useful subsets of Sn(31542). G b a

slide-55
SLIDE 55

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Subset Definitions This structure allows us to define several useful subsets of Sn(31542). G b a

slide-56
SLIDE 56

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Subset Definitions This structure allows us to define several useful subsets of Sn(31542). G b a

  • F

  c b a  

slide-57
SLIDE 57

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Subset Definitions This structure allows us to define several useful subsets of Sn(31542). G b a

  • F

  c b a  

  • G

b a

  • =
  • F

  b a  

slide-58
SLIDE 58

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for Sn(31542) For π ∈ Sn(31542), one of the following must happen j, m > 1

slide-59
SLIDE 59

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for Sn(31542) For π ∈ Sn(31542), one of the following must happen j, m > 1

  • Sn(31542)
  • =
  • Sn−1(31542)
slide-60
SLIDE 60

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for Sn(31542) For π ∈ Sn(31542), one of the following must happen j, m > 1

  • Sn(31542)
  • =
  • Sn−1(31542)
  • +

n

  • j=2

n

  • m=j

m − 2 j − 2

  • ·
  • G

n − m m − j

  • .
slide-61
SLIDE 61

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for Sn(31542) For π ∈ Sn(31542), one of the following must happen j, m > 1

  • Sn(31542)
  • =
  • Sn−1(31542)
  • +

n

  • j=2

n

  • m=j

m − 2 j − 2

  • ·
  • F

  n − m m − j  

  • .
slide-62
SLIDE 62

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  b a  

  • =

b + a a

  • ,
  • F

  c  

  • =
  • Sc(31542)
  • .
slide-63
SLIDE 63

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
slide-64
SLIDE 64

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b − 1 a − 1

slide-65
SLIDE 65

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b − 1 a − 1

  • +
  • F

  c b + a − 1  

  • ·

a + b − 1 a

slide-66
SLIDE 66

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b − 1 a − 1

  • +
  • F

  c b + a − 1  

  • ·

a + b − 1 a

  • +

c

  • ℓ=1
  • F

  c − ℓ b + ℓ − 1 a  

  • .
slide-67
SLIDE 67

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b − 1 a − 1

  • +
  • F

  c b + a − 1  

  • ·

a + b − 1 a

  • +

c

  • ℓ=1
  • F

  c − ℓ b + ℓ − 1 a  

  • .
slide-68
SLIDE 68

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b − 1 a − 1

  • +
  • F

  c b + a − 1  

  • ·

a + b − 1 a

  • +

c

  • ℓ=1
  • F

  c − ℓ b + ℓ − 1 a  

  • .
slide-69
SLIDE 69

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b − 1 a − 1

  • +
  • F

  c b + a − 1  

  • ·

a + b − 1 a

  • +

c

  • ℓ=1
  • F

  c − ℓ b + ℓ − 1 a  

  • .
slide-70
SLIDE 70

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b − 1 a − 1

  • +
  • F

  c b + a − 1  

  • ·

a + b − 1 a

  • +

c

  • ℓ=1
  • F

  c − ℓ b + ℓ − 1 a  

  • .
slide-71
SLIDE 71

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence for F   c b a   For π ∈ F   c b a  

  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b a

  • +

c

  • ℓ=1
  • F

  c − ℓ b + ℓ − 1 a  

  • .
slide-72
SLIDE 72

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Recurrence Together:

  • Sn(31542)
  • =
  • Sn−1(31542)
  • +

n

  • j=2

n

  • m=j

m − 2 j − 2

  • ·
  • F

  n − m m − j  

  • .
  • F

  c b a  

  • =
  • F

  c b + a − 1  

  • ·

a + b a

  • +

c

  • ℓ=1
  • F

  c − ℓ b + ℓ − 1 a  

  • .

Note that

  • Sn(31542)
  • satisfies a recurrence with binomial

coefficients rather than constant coefficients.

slide-73
SLIDE 73

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Conclusion Both Arnold’s composition sequence and the counting sequence for Sn(31542) are identical for at least 386 terms. Coincidence?

slide-74
SLIDE 74

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Conclusion Both Arnold’s composition sequence and the counting sequence for Sn(31542) are identical for at least 386 terms. Coincidence? Open Problems: Prove that

  • Sn(31542)
  • =

n

  • k=0
  • (n + 1 − k)k − (n − k)k

. Find a bijection between the elements of Sn(31542) and the compositions of Arnold’s construction.

slide-75
SLIDE 75

Avoiding the Pattern 31542 Lara Pudwell Barred Patterns Gaussian Polynomials and Partitions The sequence A047970 The pattern 31542 Conclusion

Compositions and Permutations n Compositions Permutations 1 2 3 4