An infinite family of inv-Wilf-equivalent permutation pairs · Justin Chan Pattern avoidance in permutations New result Shape-Wilf-equivalence Shape-inv-Wilf-equivalence Even-Wilf-equivalence

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

An infinite family of inv-Wilf-equivalentpermutation pairs

Justin Chan

Jul. 11, 2014



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21

3

1 2 3

= (231)

21

3

1 2 3

= (312)



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21

3

1 2 3

= (231)

21

3

1 2 3

= (312)



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contains

(435261) contains (231).



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contains

(435261) contains (231).



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avoids

(435261) avoids (312).



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Sn(π): Set of permutations of length n that avoid π.

S4(231) S4(321)

(1234) (1243) (1324)(1423) (1432) (2134)(2143) (3124) (3214)(4123) (4132) (4213)(4312) (4321)

(1234) (1243) (1324)(1342) (1423) (2134)(2143) (2314) (2341)(2413) (3124) (3142)(3412) (4123)



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S4(231) S4(321)

(1234) (1243) (1324)(1423) (1432) (2134)(2143) (3124) (3214)(4123) (4132) (4213)(4312) (4321)

(1234) (1243) (1324)(1342) (1423) (2134)(2143) (2314) (2341)(2413) (3124) (3142)(3412) (4123)



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S4(231) S4(321)

(1234) (1243) (1324)(1423) (1432) (2134)(2143) (3124) (3214)(4123) (4132) (4213)(4312) (4321)

(1234) (1243) (1324)(1342) (1423) (2134)(2143) (2314) (2341)(2413) (3124) (3142)(3412) (4123)



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S4(231) S4(321)

(1234) (1243) (1324)(1423) (1432) (2134)(2143) (3124) (3214)(4123) (4132) (4213)(4312) (4321)

(1234) (1243) (1324)(1342) (1423) (2134)(2143) (2314) (2341)(2413) (3124) (3142)(3412) (4123)

π |S3(π)| |S4(π)| |S5(π)| |S6(π)| |S7(π)| . . .

(231) 5 14 42 132 429 . . .

(321)



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S4(231) S4(321)

(1234) (1243) (1324)(1423) (1432) (2134)(2143) (3124) (3214)(4123) (4132) (4213)(4312) (4321)

(1234) (1243) (1324)(1342) (1423) (2134)(2143) (2314) (2341)(2413) (3124) (3142)(3412) (4123)

π |S3(π)| |S4(π)| |S5(π)| |S6(π)| |S7(π)| . . .

(231) 5 14 42 132 429 . . .

(321) 5 14 42 132 429 . . .

(231) and (321) are Wilf-equivalent: |Sn(231)| = |Sn(321)|for all n.



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S4(231) S4(321)

(1234) (1243) (1324)(1423) (1432) (2134)(2143) (3124) (3214)(4123) (4132) (4213)(4312) (4321)

(1234) (1243) (1324)(1342) (1423) (2134)(2143) (2314) (2341)(2413) (3124) (3142)(3412) (4123)

π |S3(π)| |S4(π)| |S5(π)| |S6(π)| |S7(π)| . . .

(231) 5 14 42 132 429 . . .

(321) 5 14 42 132 429 . . .

(231) and (321) are Wilf-equivalent: |Sn(231)| = |Sn(321)|for all n.



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Herbert S. Wilf (1931–2012)



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I In 2012, T. Dokos, T. Dwyer, B.P. Johnson, B.E.Sagan, and K. Selsor studied a strengthening ofWilf-equivalence involving the use of permutationstatistics (functions from the set of permutations tosome set such as the non-negative integers.)

I One such permutation statistic is inversion number.



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I In 2012, T. Dokos, T. Dwyer, B.P. Johnson, B.E.Sagan, and K. Selsor studied a strengthening ofWilf-equivalence involving the use of permutationstatistics (functions from the set of permutations tosome set such as the non-negative integers.)

I One such permutation statistic is inversion number.



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Inversion number of a permutation: We count each pair ofpositions where the value decreases from left to right.

(3142)

Count: |inv(3142) = 3



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(3142)

Count: | | |inv(3142) = 3



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(3142)

Count: | | |inv(3142) = 3



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(3142)

Count: | | |inv(3142) = 3



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(3142)

Count: | | |inv(3142) = 3



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(3142)

Count: | | |inv(3142) = 3



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(3142)

Count: | | |inv(3142) = 3



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(3142)

Count: | | |inv(3142) = 3



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inv(3142) = 3

1234

1 2 3 4



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inv(3142) = 3

1234

1 2 3 4



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S4(231) S4(321)

(1234) (1243) (1324)(1423) (1432) (2134)(2143) (3124) (3214)(4123) (4132) (4213)(4312) (4321)

(1234) (1243) (1324)(1342) (1423) (2134)(2143) (2314) (2341)(2413) (3124) (3142)(3412) (4123)

Group each of the permutations by inversion number.



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S4(231) S4(321)

(1234) (1243) (1324)(1423) (1432) (2134)(2143) (3124) (3214)(4123) (4132) (4213)(4312) (4321)

(1234) (1243) (1324)(1342) (1423) (2134)(2143) (2314) (2341)(2413) (3124) (3142)(3412) (4123)

Group each of the permutations by inversion number.



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S4(231) inv S4(321)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (1423) (2143)

(2314) (3124)

(1432) (3214) (4123)(2134)

3(2341) (2413)(3142) (4123)

(4132) (4213) 4 (3412)

(4312) 5

(4321) 6

I4(231, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4(321, q) = 1 + 3q + 5q2 + 4q3 + q4

(231) and (321) are not inv-Wilf-equivalent.



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S4(231) inv S4(321)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (1423) (2143)

(2314) (3124)

(1432) (3214) (4123)(2134)

3(2341) (2413)(3142) (4123)

(4132) (4213) 4 (3412)

(4312) 5

(4321) 6

I4(231, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4(321, q) = 1 + 3q + 5q2 + 4q3 + q4




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S4(231) inv S4(321)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (1423) (2143)

(2314) (3124)

(1432) (3214) (4123)(2134)

3(2341) (2413)(3142) (4123)

(4132) (4213) 4 (3412)

(4312) 5

(4321) 6

I4(231, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4(321, q) = 1 + 3q + 5q2 + 4q3 + q4




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S4(231) inv S4(321)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (1423) (2143)

(2314) (3124)

(1432) (3214) (4123)(2134)

3(2341) (2413)(3142) (4123)

(4132) (4213) 4 (3412)

(4312) 5

(4321) 6



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S4(231) inv S4(321)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (1423) (2143)

(2314) (3124)

(1432) (3214) (4123)(2134)

3(2341) (2413)(3142) (4123)

(4132) (4213) 4 (3412)

(4312) 5

(4321) 6



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (1423) (2143)

(2314) (3124)

(1432) (3214) (4123)(2134)

3(2341) (2413)(3142) (4123)

(4132) (4213) 4 (3412)

(4312) 5

(4321) 6



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

I4(231, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4(312, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

Similarly, In(231, q) = In(312, q) for all n.(231) and (312) are inv-Wilf-equivalent.Inv-Wilf-equivalence implies Wilf-equivalence.



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

I4(231, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4(312, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6




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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

I4(231, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4(312, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

Similarly, In(231, q) = In(312, q) for all n.

(231) and (312) are inv-Wilf-equivalent.Inv-Wilf-equivalence implies Wilf-equivalence.



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

I4(231, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4(312, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

Similarly, In(231, q) = In(312, q) for all n.(231) and (312) are inv-Wilf-equivalent.

Inv-Wilf-equivalence implies Wilf-equivalence.



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

I4(231, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4(312, q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6




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Wilf-equivalent



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inv-Wilf-equivalent

Wilf-equivalent

AAAU



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In fact, inv-Wilf-equivalence of (231) and (312) is trivial.

If a permutation avoids (231), then its diagonal reflectionavoids (312), and has the same inversion number.

1234

1 2 3 41234

1 2 3 4



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1234

1 2 3 41234

1 2 3 4



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1234

1 2 3 41234

1 2 3 4



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I Dokos et al. conjecture (2012) that allinv-Wilf-equivalences are trivial (one permutation canbe obtained from the other by reflection through eitherdiagonal or by 180-degree rotation).

I Verified computationally in the same paper for allpermutation pairs π, π′ of length ≤ 5: Sufficient toshow In(π, q) 6= In(π′, q) for a single value of n. (In thiscase, n = 8 suffices.)

I What about permutation pairs of length 6?



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It turns out that (231564) and (312564) form a nontrivialinv-Wilf-equivalent pair!

(231564) (312564)

(231)⊕ (231) (312)⊕ (231)

In fact, for every permutation γ, the pair of permutations(231)⊕ γ and (312)⊕ γ is inv-Wilf-equivalent!This results in an infinite number of pairs of nontrivialinv-Wilf-equivalent permutations.



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(231564) (312564)

(231)⊕ (231) (312)⊕ (231)




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(231564) (312564)

(231)⊕ (231) (312)⊕ (231)

In fact, for every permutation γ, the pair of permutations(231)⊕ γ and (312)⊕ γ is inv-Wilf-equivalent!

This results in an infinite number of pairs of nontrivialinv-Wilf-equivalent permutations.



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(231564) (312564)

(231)⊕ (231) (312)⊕ (231)




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I What happens when a permutation avoids (231)⊕ γ?

I Let’s take (2317456) = (231)⊕ (4123) as an example.



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I What happens when a permutation avoids (231)⊕ γ?

I Let’s take (2317456) = (231)⊕ (4123) as an example.



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avoids



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avoids



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avoids



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avoids



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avoids



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avoids



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avoids



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avoids



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avoids



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The grey region “avoids”



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The grey region “avoids”



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Cut out all rows and columns of the grey region that do nothave dots.



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“avoids”



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“avoids”



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“avoids”

Necessary and sufficient condition.



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“avoids”

Necessary and sufficient condition.



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contains

Young diagram



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contains

Transversal in Young diagram



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contains

The submatrix must be completely inside the Young diagramto count as containment.



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contains




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avoids




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avoids




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avoids



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avoids



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avoids



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“avoids”



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SY (π): Set of transversals in Young diagram Y that avoid π.

Y |SY (123)| |SY (321)|

5 5

10 10

13 13

14 14

In fact, |SY (123)| = |SY (321)| for all Young diagrams Y .(123) and (321) are shape-Wilf-equivalent.Shape-Wilf-equivalence implies Wilf-equivalence.



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Y |SY (123)| |SY (321)|

5 5

10 10

13 13

14 14




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Y |SY (123)| |SY (321)|

5 5

10 10

13 13

14 14

In fact, |SY (123)| = |SY (321)| for all Young diagrams Y .

(123) and (321) are shape-Wilf-equivalent.Shape-Wilf-equivalence implies Wilf-equivalence.



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Y |SY (123)| |SY (321)|

5 5

10 10

13 13

14 14

In fact, |SY (123)| = |SY (321)| for all Young diagrams Y .(123) and (321) are shape-Wilf-equivalent.

Shape-Wilf-equivalence implies Wilf-equivalence.



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Y |SY (123)| |SY (321)|

5 5

10 10

13 13

14 14




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inv-Wilf-equivalent

Wilf-equivalent

AAAU



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inv-Wilf-equivalent shape-Wilf-equivalent

Wilf-equivalent

��

AAAU



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Wilf-equivalent

(1342), (1423) (123), (321)

��

AAAU



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I Stankova, West (2002): (231) and (312) areshape-Wilf-equivalent.

I Backelin, West, Xin (2007): (123 . . . n) and (n . . . 321)are shape-Wilf-equivalent for all n ≥ 2.

I Backelin, West, Xin (2007): If α and β areshape-Wilf-equivalent, then so are α⊕ γ and β ⊕ γ forall permutations γ.

So (231)⊕ γ and (312)⊕ γ are shape-Wilf-equivalent for allpermutations γ.How do we show that they are inv-Wilf-equivalent?



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So (231)⊕ γ and (312)⊕ γ are shape-Wilf-equivalent for allpermutations γ.

How do we show that they are inv-Wilf-equivalent?



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Wilf-equivalent

��

AAAU



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shape-inv-Wilf-equivalent


Wilf-equivalent

��

AAAU

AAAU

��



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Combine shape-Wilf-equivalence and inv-Wilf-equivalence todefine shape-inv-Wilf-equivalence: For each Youngdiagram Y , the sets of transversals in Y avoiding each ofthe two permutations have the same distribution accordingto inversion number.

To show that (231)⊕ γ and (312)⊕ γ are inv-Wilf-equivalentfor all permutations γ, we just need the following.



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Combine shape-Wilf-equivalence and inv-Wilf-equivalence todefine shape-inv-Wilf-equivalence: For each Youngdiagram Y , the sets of transversals in Y avoiding each ofthe two permutations have the same distribution accordingto inversion number.To show that (231)⊕ γ and (312)⊕ γ are inv-Wilf-equivalentfor all permutations γ, we just need the following.



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I Prove that if α and β are shape-inv-Wilf-equivalent,then so are α⊕ γ and β ⊕ γ.

I Prove that (231) and (312) areshape-inv-Wilf-equivalent.



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I Extend Backelin-West-Xin result onshape-Wilf-equivalence of α⊕ γ and β ⊕ γ.


I Extend Stankova-West result on shape-Wilf-equivalenceof (231) and (312).

I I (Y ): inv-polynomial (generating function) fortransversals in Young diagram Y that avoid (231).

I A functional relation called row decomposition(expressing I (Y ) in terms of “smaller” I (Z )).

I The “reflection” of row decomposition called columndecomposition (prove by induction).

I Together, these show that (231) and (312) areshape-inv-Wilf-equivalent.



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Wilf-equivalent

��

AAAU

AAAU

��



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Wilf-equivalent

(231), (312)

��

AAAU

AAAU

��



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Wilf-equivalent

(231), (312)

(231), (312)(231)⊕ γ, (312)⊕ γ

(231)⊕ γ, (312)⊕ γ

��

AAAU

AAAU

��



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We now define even-Wilf-equivalence, introduced by Baxterand Jaggard in 2011.



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

Red denotes even permutation. (A permutation is even ifand only if its inversion number is even.)



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

Red denotes even permutation. (A permutation is even ifand only if its inversion number is even.)



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

An(π): Set of even permutations of length n that avoid π.

|A4(231)| = |A4(312)| = 7, and similarly|An(231)| = |An(312)| for all n.(231) and (312) are even-Wilf-equivalent.Inv-Wilf-equivalence implies even-Wilf-equivalence.



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S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

An(π): Set of even permutations of length n that avoid π.|A4(231)| = |A4(312)| = 7, and similarly|An(231)| = |An(312)| for all n.

(231) and (312) are even-Wilf-equivalent.Inv-Wilf-equivalence implies even-Wilf-equivalence.



Justin Chan


New result




S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

An(π): Set of even permutations of length n that avoid π.|A4(231)| = |A4(312)| = 7, and similarly|An(231)| = |An(312)| for all n.(231) and (312) are even-Wilf-equivalent.

Inv-Wilf-equivalence implies even-Wilf-equivalence.



Justin Chan


New result




S4(231) inv S4(312)

(1234) 0 (1234)

(1243) (1324) (2134) 1 (1243) (1324) (2134)

(1423) (2143) (3124)(2134)

2(1342) (2143) (2314)

(2134)

(1432) (3214) (4123)(2134)

3(1432) (3214) (2341)

(2134)

(4132) (4213) 4 (2431) (3241)

(4312) 5 (3421)

(4321) 6 (4321)

An(π): Set of even permutations of length n that avoid π.|A4(231)| = |A4(312)| = 7, and similarly|An(231)| = |An(312)| for all n.(231) and (312) are even-Wilf-equivalent.Inv-Wilf-equivalence implies even-Wilf-equivalence.



Justin Chan


New result





inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalent

(231), (312)

?

HHHj

HHHj ?



Justin Chan


New result





inv-Wilf-equivalent


Wilf-equivalenteven-Wilf-equivalent

(231), (312)

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HHHj

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Justin Chan


New result





inv-Wilf-equivalent



(231), (312)

?Baxter, Jaggard (2011)

-

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HHHj

HHHj ?��



Justin Chan


New result




I Baxter and Jaggard determined all pairs of permutationsof length 4 or less which are even-Wilf-equivalent.

I They also determined which pairs of length 6 areeven-Wilf-equivalent, except for the pairs(231564), (312564) and (465132), (465213)(conjectured both are).

I Proving these even-Wilf-equivalences would completethe classification for length 6.



Justin Chan


New result









Justin Chan


New result









Justin Chan


New result





inv-Wilf-equivalent



(231), (312)


-

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Justin Chan


New result





inv-Wilf-equivalent



(231), (312)

(231564), (312564)?(465132), (465213)?

Baxter, Jaggard (2011)


-

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HHHj

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Justin Chan


New result




AY (π): Set of transversals with even inversion number inYoung diagram Y that avoid π.

Y |AY (213)| |AY (321)|

2 2

5 5

6 6

7 7

In fact, |AY (213)| = |AY (321)| for all Young diagrams Y .(213) and (321) are even-shape-Wilf-equivalent.Even-shape-Wilf-equivalence implies even-Wilf-equivalence.



Justin Chan


New result





Y |AY (213)| |AY (321)|

2 2

5 5

6 6

7 7




Justin Chan


New result





Y |AY (213)| |AY (321)|

2 2

5 5

6 6

7 7

In fact, |AY (213)| = |AY (321)| for all Young diagrams Y .

(213) and (321) are even-shape-Wilf-equivalent.Even-shape-Wilf-equivalence implies even-Wilf-equivalence.



Justin Chan


New result





Y |AY (213)| |AY (321)|

2 2

5 5

6 6

7 7

In fact, |AY (213)| = |AY (321)| for all Young diagrams Y .(213) and (321) are even-shape-Wilf-equivalent.

Even-shape-Wilf-equivalence implies even-Wilf-equivalence.



Justin Chan


New result





Y |AY (213)| |AY (321)|

2 2

5 5

6 6

7 7




Justin Chan


New result





inv-Wilf-equivalent



(231), (312)

(231564), (312564)?(465132), (465213)?



-

?

HHHj

HHHj ?��



Justin Chan


New result





inv-Wilf-equivalent



even-shape-Wilf-equivalent

(231), (312)

(231564), (312564)?(465132), (465213)?



-

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HHHj

HHHj ?��?



Justin Chan


New result





inv-Wilf-equivalent




(231), (312)

(231564), (312564)?(465132), (465213)?


(231), (312)?Baxter, Jaggard (2011)


-

?

HHHj

HHHj ?��?



Justin Chan


New result





inv-Wilf-equivalent




(231), (312)

(231564), (312564)?(465132), (465213)?


(231), (312)?Baxter, Jaggard (2011)


-

?

HHHj

HHHj ?��?

��



Justin Chan


New result





inv-Wilf-equivalent




(231), (312)

(231564), (312564)?(465132), (465213)?



-

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HHHj

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��



Justin Chan


New result





inv-Wilf-equivalent




(231), (312)

(231564), (312564)?(465132), (465213)?


(231), (312)


-

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HHHj

HHHj ?��?

��



Justin Chan


New result





inv-Wilf-equivalent




(231), (312)

(231), (312)


-

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HHHj

HHHj ?��?

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Justin Chan


New result





inv-Wilf-equivalent




(231), (312)

(231564), (312564)

(465132), (465213)

(231), (312)


-

?

HHHj

HHHj ?��?

��



Justin Chan


New result





inv-Wilf-equivalent




(231), (312)

(231564), (312564)

(465132), (465213)

(231), (312)


-

? -

?

HHHj

HHHj ?��?

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Justin Chan


New result




End.

Documents

An infinite family of inv-Wilf-equivalent permutation pairs · Justin Chan Pattern avoidance in permutations New result Shape-Wilf-equivalence Shape-inv-Wilf-equivalence Even-Wilf-equivalence