Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter...
Transcript of Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter...
![Page 1: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/1.jpg)
Baxter permutations and meanders
Eric Fusy (LIX, Ecole Polytechnique)
Journees Viennot, 28-29 juin 2012, Labri
![Page 2: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/2.jpg)
Meanders on two lines• A 2-line meander
![Page 3: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/3.jpg)
Meanders on two lines• A 2-line meander encoded by a permutation
1 234 567 8 9
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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
![Page 5: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/5.jpg)
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
![Page 6: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/6.jpg)
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
![Page 7: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/7.jpg)
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
![Page 8: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/8.jpg)
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
![Page 9: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/9.jpg)
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
![Page 10: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/10.jpg)
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
![Page 11: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/11.jpg)
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
![Page 12: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/12.jpg)
Permutations for monotone 2-line meanders
left orright?
[Baxter’64, Boyce’67&’81]
![Page 13: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/13.jpg)
Permutations for monotone 2-line meanders
left
[Baxter’64, Boyce’67&’81]
![Page 14: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/14.jpg)
Permutations for monotone 2-line meanders
then has togo left
left`
[Baxter’64, Boyce’67&’81]
![Page 15: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/15.jpg)
Permutations for monotone 2-line meanders
``
[Baxter’64, Boyce’67&’81]
![Page 16: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/16.jpg)
Permutations for monotone 2-line meanders
``
left orright?
[Baxter’64, Boyce’67&’81]
![Page 17: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/17.jpg)
Permutations for monotone 2-line meanders
``
left orright?
[Baxter’64, Boyce’67&’81]
![Page 18: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/18.jpg)
Permutations for monotone 2-line meanders
``
left orright?
r
then has togo right
[Baxter’64, Boyce’67&’81]
![Page 19: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/19.jpg)
Permutations for monotone 2-line meanders
``
left orright?
r r
[Baxter’64, Boyce’67&’81]
![Page 20: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/20.jpg)
Permutations for monotone 2-line meanders
``
left orright?
r r
`
[Baxter’64, Boyce’67&’81]
![Page 21: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/21.jpg)
Permutations for monotone 2-line meanders
``
left orright?
r r
`
then has togo left
[Baxter’64, Boyce’67&’81]
![Page 22: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/22.jpg)
Permutations for monotone 2-line meanders
``
left orright?
r r
``
[Baxter’64, Boyce’67&’81]
![Page 23: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/23.jpg)
Permutations for monotone 2-line meanders
``
left orright?
r r
``
[Baxter’64, Boyce’67&’81]
![Page 24: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/24.jpg)
Permutations for monotone 2-line meanders
``
left orright?
r r
``
white points are either:
or
[Baxter’64, Boyce’67&’81]
rising descending
![Page 25: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/25.jpg)
Permutations for monotone 2-line meanders
``
left orright?
r r
``
white points are either:
or
[Baxter’64, Boyce’67&’81]
rising descending
![Page 26: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/26.jpg)
Permutations for monotone 2-line meanders
``
r r
``
white points are either:
or
[Baxter’64, Boyce’67&’81]
rising descending
![Page 27: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/27.jpg)
Permutations for monotone 2-line meanders[Baxter’64, Boyce’67&’81]
``
r r
``
white points are either:
rising descending
or
![Page 28: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/28.jpg)
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
![Page 29: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/29.jpg)
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
![Page 30: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/30.jpg)
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
or
![Page 31: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/31.jpg)
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
1) Draw the blue curve
or
![Page 32: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/32.jpg)
Inverse constructionFrom a complete Baxter permutation to a monotone 2-line meander
white points are either:
1) Draw the blue curve
or
![Page 33: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/33.jpg)
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
![Page 34: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/34.jpg)
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
![Page 35: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/35.jpg)
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
![Page 36: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/36.jpg)
Complete and reduced Baxter permutations
complete reduced
![Page 37: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/37.jpg)
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
![Page 38: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/38.jpg)
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a descent
![Page 39: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/39.jpg)
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a descent
![Page 40: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/40.jpg)
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a descent
![Page 41: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/41.jpg)
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a rise
![Page 42: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/42.jpg)
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a rise
![Page 43: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/43.jpg)
Complete and reduced Baxter permutations
complete reduced
• complete one can be recovered from reduced one
case of a rise
![Page 44: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/44.jpg)
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
![Page 45: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/45.jpg)
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
![Page 46: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/46.jpg)
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
)
![Page 47: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/47.jpg)
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 ⇔
![Page 48: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/48.jpg)
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 ⇔
![Page 49: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/49.jpg)
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 ⇔
![Page 50: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/50.jpg)
Local conditions for monotone 2-line meanders
![Page 51: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/51.jpg)
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
![Page 52: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/52.jpg)
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
![Page 53: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/53.jpg)
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
![Page 54: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/54.jpg)
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
![Page 55: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/55.jpg)
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
![Page 56: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/56.jpg)
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
![Page 57: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/57.jpg)
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
![Page 58: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/58.jpg)
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
![Page 59: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/59.jpg)
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
![Page 60: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/60.jpg)
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
![Page 61: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/61.jpg)
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
![Page 62: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/62.jpg)
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
![Page 63: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/63.jpg)
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
![Page 64: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/64.jpg)
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
![Page 65: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/65.jpg)
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
![Page 66: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/66.jpg)
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
![Page 67: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/67.jpg)
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
![Page 68: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/68.jpg)
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
![Page 69: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/69.jpg)
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
![Page 70: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/70.jpg)
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
![Page 71: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/71.jpg)
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
![Page 72: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/72.jpg)
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
![Page 73: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/73.jpg)
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
![Page 74: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/74.jpg)
Encoding a monotone 2-line meander
![Page 75: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/75.jpg)
Encoding a monotone 2-line meander
⇓
![Page 76: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/76.jpg)
Encoding a monotone 2-line meander
⇓
⇓
![Page 77: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/77.jpg)
Encoding a monotone 2-line meander
⇓
⇓
0 1 1 0 1 1
1 0 1 1 1 0
0 1 1 1 0 1
![Page 78: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/78.jpg)
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
![Page 79: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/79.jpg)
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
![Page 80: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/80.jpg)
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]
![Page 81: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/81.jpg)
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)
![Page 82: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/82.jpg)
Enumeration using the LGV lemmar
each path haslength n− 1
A1
A2
A3
B1
B2
B3
![Page 83: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/83.jpg)
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
)
![Page 84: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/84.jpg)
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
![Page 85: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/85.jpg)
Alternating (reduced) Baxter permutations
1) Case n even, n = 2k[Cori-Dulucq-Viennot’86], [Dulucq-Guibert’98]
π= 1 23498 67 510
![Page 86: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/86.jpg)
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
![Page 87: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/87.jpg)
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
![Page 88: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/88.jpg)
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
![Page 89: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/89.jpg)
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
![Page 90: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/90.jpg)
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
![Page 91: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/91.jpg)
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
![Page 92: Eric Fusy (LIX, Baxter permutations and meandersfusy/Talks/baxter_meanders.pdf · Baxter permutations and meanders Eric Fusy (LIX, Ecole Polytechnique) Journ ees Viennot, 28-29 juin](https://reader034.fdocuments.us/reader034/viewer/2022042917/5f5b283503163e3ef43cf550/html5/thumbnails/92.jpg)
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