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 family of inv-Wilf-equivalent permutation pairs Justin Chan Jul. 11, 2014
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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
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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.
(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.
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.
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.
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.