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Baxter permutations and meanders
Eric Fusy (LIX, Ecole Polytechnique)
Journees Viennot, 28-29 juin 2012, Labri
Meanders on two lines• A 2-line meander
Meanders on two lines• A 2-line meander encoded by a permutation
1 234 567 8 9
Meanders on two lines• A 2-line meander encoded by a permutation
• Monotone 2-line meander:can be obtained from two monotone lines (one in x, the other in y)
1 234 567 8 9
Meanders on two lines• A 2-line meander encoded by a permutation
• Monotone 2-line meander:can be obtained from two monotone lines (one in x, the other in y)
associatedpermutation
1 234 567 8 9
Meanders on two lines• A 2-line meander encoded by a permutation
• Monotone 2-line meander:can be obtained from two monotone lines (one in x, the other in y)
associatedpermutation
Which permutations can be obtained this way ?
1 234 567 8 9
Meanders on two lines• A 2-line meander encoded by a permutation
• Monotone 2-line meander:can be obtained from two monotone lines (one in x, the other in y)
associatedpermutation
Which permutations can be obtained this way ?
1 234 567 8 9
Meanders on two lines• A 2-line meander encoded by a permutation
• Monotone 2-line meander:can be obtained from two monotone lines (one in x, the other in y)
associatedpermutation
Which permutations can be obtained this way ?Maps odd numbers to odd numbers, even numbers to even numbers
1 234 567 8 9
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
left orright?
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
left
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
then has togo left
left`
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
r
then has togo right
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
r r
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
r r
`
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
r r
`
then has togo left
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
r r
``
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
r r
``
[Baxter’64, Boyce’67&’81]
Permutations for monotone 2-line meanders
``
left orright?
r r
``
white points are either:
or
[Baxter’64, Boyce’67&’81]
rising descending
Permutations for monotone 2-line meanders
``
left orright?
r r
``
white points are either:
or
[Baxter’64, Boyce’67&’81]
rising descending
Permutations for monotone 2-line meanders
``
r r
``
white points are either:
or
[Baxter’64, Boyce’67&’81]
rising descending
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
``
r r
``
white points are either:
rising descending
or
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
``
r r
``
white points are either:
Permutations mapping even to even, odd to odd, and satisfying condition
rising descending
or
shown on the right are called complete Baxter permutations
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
``
r r
``
white points are either:
Permutations mapping even to even, odd to odd, and satisfying condition
Theorem ([Boyce’81] reformulated bijectively):Monotone 2-line meanders with 2n− 1 crossings are in bijection withcomplete Baxter permutations on 2n− 1 elements
rising descending
or
shown on the right are called complete Baxter permutations
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
or
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
1) Draw the blue curve
or
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
1) Draw the blue curve
or
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
1) Draw the blue curve
2) Draw the red curve
or
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
1) Draw the blue curve
2) Draw the red curve
or
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
1) Draw the blue curve
2) Draw the red curve
The two curves meet only at the permutation points(because of the empty area-property at white points)
or
Complete and reduced Baxter permutations
complete reduced
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a descent
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a descent
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a descent
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a rise
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a rise
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a rise
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one• reduced one is characterized by forbidden patterns
2− 41− 3 and 3− 14− 2
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one• reduced one is characterized by forbidden patterns
2− 41− 3 and 3− 14− 2• permutation on white points (called anti-Baxter)is characterized by forbidden patterns
2− 14− 3 and 3− 41− 2
Counting results• Baxter permutations
[Chung et al’78] [Mallows’79]
- Number of reduced Baxter permutations with n elements
bn =
n−1∑r=0
2
n(n+ 1)2
(n+ 1
r
)(n+ 1
r + 1
)(n+ 1
r + 2
)
Counting results• Baxter permutations
[Chung et al’78] [Mallows’79]
- Number of reduced Baxter permutations with n elements
bn =
n−1∑r=0
2
n(n+ 1)2
(n+ 1
r
)(n+ 1
r + 1
)(n+ 1
r + 2
)
- Bijective proof: [Viennot’81], [Dulucq-Guibert’98]
5 7 6 2 1 4 3 ⇔
Counting results• Baxter permutations
[Chung et al’78] [Mallows’79]
- Number of reduced Baxter permutations with n elements
bn =
n−1∑r=0
2
n(n+ 1)2
(n+ 1
r
)(n+ 1
r + 1
)(n+ 1
r + 2
)
- Bijective proof: [Viennot’81], [Dulucq-Guibert’98]
• Subfamilies- alternating [Cori-Dulucq-Viennot’86], [Dulucq-Guibert’98]
- doubly alternating [Guibert-Linusson’00]CatkCatk if n = 2k CatkCatk+1 if n = 2k + 1
Catk where k = bn/2c
5 7 6 2 1 4 3 ⇔
Counting results• Baxter permutations
[Chung et al’78] [Mallows’79]
- Number of reduced Baxter permutations with n elements
bn =
n−1∑r=0
2
n(n+ 1)2
(n+ 1
r
)(n+ 1
r + 1
)(n+ 1
r + 2
)
- Bijective proof: [Viennot’81], [Dulucq-Guibert’98]
• Subfamilies- alternating [Cori-Dulucq-Viennot’86], [Dulucq-Guibert’98]
- doubly alternating [Guibert-Linusson’00]
• anti-Baxter permutations
CatkCatk if n = 2k CatkCatk+1 if n = 2k + 1
Catk where k = bn/2c
an =
b(n+1)/2c∑i=0
(−1)i(n+ 1− i
i
)bn+1−i[Asinowski et al’10]
5 7 6 2 1 4 3 ⇔
Local conditions for monotone 2-line meanders
Local conditions for monotone 2-line meanders
Conditions- two (bipartite) matchings missing a (black) point
(one matching above, one below the blue line)
⇔
- white points are either or
rising descending
Local conditions for monotone 2-line meanders
Conditions- two (bipartite) matchings missing a (black) point
(one matching above, one below the blue line)
⇔
- white points are either or
Proof of ⇐Assume there is a red loop (say, clockwise):
rising descending
Local conditions for monotone 2-line meanders
Conditions- two (bipartite) matchings missing a (black) point
(one matching above, one below the blue line)
⇔
- white points are either or
Proof of ⇐Assume there is a red loop (say, clockwise):
then the leftmost and therighmost point on the loopare of different colors
rising descending
Local conditions for monotone 2-line meanders
Conditions- two (bipartite) matchings missing a (black) point
(one matching above, one below the blue line)
⇔
- white points are either or
Proof of ⇐Assume there is a red loop (say, clockwise):
then the leftmost and therighmost point on the loopare of different colors
impossible
rising descending
Local conditions for monotone 2-line meanders
Conditions- two (bipartite) matchings missing a (black) point
(one matching above, one below the blue line)
⇔
- white points are either or
Proof of ⇐Assume there is a red loop (say, clockwise):
then the leftmost and therighmost point on the loopare of different colors
impossible
⇒ we have a 2-line meander
rising descending
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
1
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
12
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
By similar argument as toshow there is no red loop
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
ii+1 ︸︷︷︸already labelled
By similar argument as toshow there is no red loop
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
5
ii+1 ︸︷︷︸already labelled
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
7
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
78
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
789
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
789 10
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
789 10 11
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
789 10 1112
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
789 10 111213
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
789 10 111213
- two (bipartite) matchings missing a (black) point
Local conditions for monotone 2-line meanders
Conditions
(one matching above, one below the blue line)
⇔
- white nodes are either or
Proof of ⇐: construct permutation step by step
123 4
important observation:
i i+1︸︷︷︸already labelled
56
ii+1 ︸︷︷︸already labelled
789 10 111213
Similarly:
or
- two (bipartite) matchings missing a (black) point
Encoding a monotone 2-line meander
Encoding a monotone 2-line meander
⇓
Encoding a monotone 2-line meander
⇓
⇓
Encoding a monotone 2-line meander
⇓
⇓
0 1 1 0 1 1
1 0 1 1 1 0
0 1 1 1 0 1
Encoding a monotone 2-line meander
⇓
⇓
0 1 1 0 1 1
1 0 1 1 1 0
0 1 1 1 0 1 ⇒0
1
Encoding a monotone 2-line meander
⇓
⇓
0 1 1 0 1 1
1 0 1 1 1 0
0 1 1 1 0 1 ⇒0
1
# rises
each path haslength n− 1
Encoding a monotone 2-line meander
⇓
⇓
0 1 1 0 1 1
1 0 1 1 1 0
0 1 1 1 0 1 ⇒0
1
# rises
each path haslength n− 1
close to encoding in [Viennot’81,Dulucq-Guibert’98]
Encoding a monotone 2-line meander
⇓
⇓
0 1 1 0 1 1
1 0 1 1 1 0
0 1 1 1 0 1 ⇒0
1
# rises
each path haslength n− 1
close to encoding in [Viennot’81,Dulucq-Guibert’98]exactly coincides with encoding in [Felsner-F-Noy-Orden’11](uses “equatorial line” in separating decompositions of quadrangulations)
Enumeration using the LGV lemmar
each path haslength n− 1
A1
A2
A3
B1
B2
B3
Enumeration using the LGV lemmar
each path haslength n− 1
A1
A2
A3
B1
B2
B3
Let ai,j = # (upright lattice paths from Ai to Bj) =(
n−1x(Bj)−x(Ai)
)By the Lindstroem-Gessel-Viennot Lemma (used in [Viennot’81])
the number bn,r of such nonintersecting triples of paths is
bn,r = Det(ai,j) =
∣∣∣∣∣∣∣∣(n−1r
) (n−1r+1
) (n−1r+2
)(n−1r−1) (
n−1r
) (n−1r+1
)(n−1r−2) (
n−1r−1) (
n−1r
)∣∣∣∣∣∣∣∣=
2n(n+1)2
(n+1r
)(n+1r+1
)(n+1r+2
)
Enumeration using the LGV lemmar
each path haslength n− 1
A1
A2
A3
B1
B2
B3
Let ai,j = # (upright lattice paths from Ai to Bj) =(
n−1x(Bj)−x(Ai)
)By the Lindstroem-Gessel-Viennot Lemma (used in [Viennot’81])
the number bn,r of such nonintersecting triples of paths is
bn,r = Det(ai,j) =
∣∣∣∣∣∣∣∣(n−1r
) (n−1r+1
) (n−1r+2
)(n−1r−1) (
n−1r
) (n−1r+1
)(n−1r−2) (
n−1r−1) (
n−1r
)∣∣∣∣∣∣∣∣=
2n(n+1)2
(n+1r
)(n+1r+1
)(n+1r+2
)
bn,r is also the number of reduced Baxter permutations of sizen with r rises
Alternating (reduced) Baxter permutations
1) Case n even, n = 2k[Cori-Dulucq-Viennot’86], [Dulucq-Guibert’98]
π= 1 23498 67 510
Alternating (reduced) Baxter permutations
alternation ⇔ middle word is 010101 . . . 0
1) Case n even, n = 2k[Cori-Dulucq-Viennot’86], [Dulucq-Guibert’98]
π= 1 23498 67 510
Alternating (reduced) Baxter permutations
alternation ⇔ middle word is 010101 . . . 0
1) Case n even, n = 2k
middle path is
[Cori-Dulucq-Viennot’86], [Dulucq-Guibert’98]
π= 1 23498 67 510
Alternating (reduced) Baxter permutations
alternation ⇔ middle word is 010101 . . . 0
1) Case n even, n = 2k
middle path is
⇒
[Cori-Dulucq-Viennot’86], [Dulucq-Guibert’98]
There are CatkCatk alternating(reduced) Baxter permutations of size n
k
π= 1 23498 67 510
Alternating (reduced) Baxter permutations
alternation ⇔ middle word is 010101 . . . 1
2) Case n odd, n = 2k + 1
middle path is
⇒
[Cori-Dulucq-Viennot’86], [Dulucq-Guibert’98]
There are CatkCatk+1 alternating(reduced) Baxter permutations of size n
k
π= 12 3 49 8 6 7 5
k
Doubly alternating (reduced) Baxter permutations[Guibert-Linusson’00]1) Case n even, n = 2k
1 2438675π =
alternation of π: middle word is 010101 . . . 0
alternation of π−1: black points are or
middle path is
Doubly alternating (reduced) Baxter permutations[Guibert-Linusson’00]1) Case n even, n = 2k
1 2438675π =
alternation of π: middle word is 010101 . . . 0
alternation of π−1: black points are or
⇒ There are Catk doubly alternating(reduced) Baxter permutations of size nmirror
of eachother
middle path is
k
Doubly alternating (reduced) Baxter permutations[Guibert-Linusson’00]2) Case n odd, n = 2k + 1
1 2 43 8 675π=
alternation of π: middle word is 010101 . . . 1alternation of π−1: black points are or
⇒ There are Catk doubly alternating(reduced) Baxter permutations of size nmirror
of eachother
9
not intop-word
change of convention
not inbottom-word
k