An infinite family of inv-Wilf-equivalent permutation pairs · Justin Chan Pattern avoidance in...

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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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

21

3

1 2 3

= (231)

21

3

1 2 3

= (312)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

21

3

1 2 3

= (231)

21

3

1 2 3

= (312)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

contains

(435261) contains (231).

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

contains

(435261) contains (231).

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

(435261) avoids (312).

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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)

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

(231) 5 14 42 132 429 . . .

(321)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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)

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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)

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

Herbert S. Wilf (1931–2012)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

(3142)

Count: |inv(3142) = 3

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

(3142)

Count: | | |inv(3142) = 3

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

(3142)

Count: | | |inv(3142) = 3

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

(3142)

Count: | | |inv(3142) = 3

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

(3142)

Count: | | |inv(3142) = 3

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

(3142)

Count: | | |inv(3142) = 3

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

(3142)

Count: | | |inv(3142) = 3

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

(3142)

Count: | | |inv(3142) = 3

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

inv(3142) = 3

1234

1 2 3 4

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

inv(3142) = 3

1234

1 2 3 4

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

Wilf-equivalent

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

inv-Wilf-equivalent

Wilf-equivalent

AAAU

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

The grey region “avoids”

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

The grey region “avoids”

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

“avoids”

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

“avoids”

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

“avoids”

Necessary and sufficient condition.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

“avoids”

Necessary and sufficient condition.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

contains

Young diagram

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

contains

Transversal in Young diagram

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

contains

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

contains

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

avoids

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

“avoids”

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

inv-Wilf-equivalent

Wilf-equivalent

AAAU

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

inv-Wilf-equivalent shape-Wilf-equivalent

Wilf-equivalent

����

AAAU

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

inv-Wilf-equivalent shape-Wilf-equivalent

Wilf-equivalent

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

����

AAAU

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

inv-Wilf-equivalent shape-Wilf-equivalent

Wilf-equivalent

����

AAAU

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent shape-Wilf-equivalent

Wilf-equivalent

����

AAAU

AAAU

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

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

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

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

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

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

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

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

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

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

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

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

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

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

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

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

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

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent shape-Wilf-equivalent

Wilf-equivalent

����

AAAU

AAAU

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent shape-Wilf-equivalent

Wilf-equivalent

(231), (312)

����

AAAU

AAAU

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent shape-Wilf-equivalent

Wilf-equivalent

(231), (312)

(231), (312)(231)⊕ γ, (312)⊕ γ

(231)⊕ γ, (312)⊕ γ

����

AAAU

AAAU

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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 infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.)

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalent

(231), (312)

?

HHHj

HHHj ?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

(231), (312)

?

HHHj

HHHj ?����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

(231), (312)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

(231), (312)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

(231), (312)

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

Baxter, Jaggard (2011)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

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.

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

(231), (312)

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

Baxter, Jaggard (2011)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

even-shape-Wilf-equivalent

(231), (312)

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

Baxter, Jaggard (2011)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

even-shape-Wilf-equivalent

(231), (312)

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

Baxter, Jaggard (2011)

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

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����?

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

even-shape-Wilf-equivalent

(231), (312)

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

Baxter, Jaggard (2011)

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

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����?

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

even-shape-Wilf-equivalent

(231), (312)

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

Baxter, Jaggard (2011)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����?

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

even-shape-Wilf-equivalent

(231), (312)

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

Baxter, Jaggard (2011)

(231), (312)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����?

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

even-shape-Wilf-equivalent

(231), (312)

(231), (312)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����?

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

even-shape-Wilf-equivalent

(231), (312)

(231564), (312564)

(465132), (465213)

(231), (312)

?Baxter, Jaggard (2011)

-

?

HHHj

HHHj ?����?

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

shape-inv-Wilf-equivalent

inv-Wilf-equivalent

shape-Wilf-equivalent

Wilf-equivalenteven-Wilf-equivalent

even-shape-Wilf-equivalent

(231), (312)

(231564), (312564)

(465132), (465213)

(231), (312)

?Baxter, Jaggard (2011)

-

? -

?

HHHj

HHHj ?����?

����

An infinite familyof

inv-Wilf-equivalentpermutation pairs

Justin Chan

Pattern avoidancein permutations

New result

Shape-Wilf-equivalence

Shape-inv-Wilf-equivalence

Even-Wilf-equivalence

End.

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