MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
Vincent Pilaud – Viviane PonsCNRS & Ecole Polytechnique – LRI, Univ. Paris-Sud
Permutrees
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
permutations binary trees binary sequences
Com
bin
ator
ics
4321
4231 43123421
34123241 2431 4213 4132
1234
1324 12432134
21432314 3124 1342 1423
3142 2413 4123 14323214 2341
−−−
−+− +−−−−+
−++ +−+ ++−
+++
Geo
met
ry
341234214321 4312
2413
4213
3214
14231432
1342
1243 12342134
1324
2341
2431
31242314
−−−
−+−
+−−
−−+
−++
+−+
++−
+++
Alg
ebra
Malvenuto-Reutenauer algebra Loday-Ronco algebra Solomon algebraFQSym = vect 〈 Fτ | τ ∈ S 〉 PBT = vect 〈 PT | T ∈ BT 〉 Rec = vect 〈 Xη | η ∈ ±∗ 〉
Fτ · Fτ ′ =∑
σ∈τ � τ ′Fσ PT · PT ′ =
∑T↗
T ′ ≤ T ′′ ≤T↖T ′
PT ′′ Xη · Xη′ = Xη+η′ + Xη−η′
4Fσ =∑
σ∈τ?τ ′Fτ ⊗ Fτ ′ 4Fγ =
∑γ cut
B(T , γ)⊗ A(T , γ) 4Xη =∑γ cut
B(η, γ)⊗ A(η, γ)
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree Recipe
I Take a word in { , , , }n
I Take a permutation
I Do the insertion: get a Leveled Permutree (bijection)
I Remove the levels: get a Permutree (surjection)
Examplen ←→ permutations of [n]n ←→ standard binary search trees
{ , }n ←→ Cambrian treesn ←→ binary sequences
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree Recipe
I Take a word in { , , , }n
I Take a permutation
I Do the insertion: get a Leveled Permutree (bijection)
I Remove the levels: get a Permutree (surjection)
Examplen ←→ permutations of [n]n ←→ standard binary search trees
{ , }n ←→ Cambrian treesn ←→ binary sequences
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:
Decorated permutation: 2751346
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
1234567
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
1234567
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
1234567
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
1234567
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
1234567
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
3
1234567
7
64
42
1
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
1234567
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
1234567
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation: 2751346Decoration:Decorated permutation: 2751346
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation : 2751346
Decorations:
Decorated Permutations:2751346 2751346 2751346 2751346Leveled permutrees:
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation : 2751346
Decorations:
Decorated Permutations:2751346 2751346 2751346 2751346Leveled permutrees:
1234567
1
2
34
5
6
7
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation : 2751346
Decorations:
Decorated Permutations:2751346 2751346 2751346 2751346Leveled permutrees:
1234567
1
2
34
5
6
7
1234567
7641 2 3 5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation : 2751346
Decorations:
Decorated Permutations:2751346 2751346 2751346 2751346Leveled permutrees:
1234567
1
2
34
5
6
7
1234567
7641 2 3 5
1234567
4
4
3
3
5
5
6
6
7
7
2
2
1
1
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation : 2751346
Decorations:
Decorated Permutations:2751346 2751346 2751346 2751346Leveled permutrees:
1234567
1
2
34
5
6
7
1234567
7641 2 3 5
1234567
4
4
3
3
5
5
6
6
7
7
2
2
1
1
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation : 2751346
Decorations:
Decorated Permutations:2751346 2751346 2751346 2751346Leveled permutrees:
1234567
1
2
34
5
6
7
1234567
7641 2 3 5
1234567
4
4
3
3
5
5
6
6
7
7
2
2
1
1
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation : 2751346
Decorations:
Decorated Permutations:2751346 2751346 2751346 2751346Leveled permutrees:
1234567
1
2
34
5
6
7
1234567
7641 2 3 5
1234567
4
4
3
3
5
5
6
6
7
7
2
2
1
1
1234567
7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
The Permutree insertionPermutation : 2751346
Decorations:
Decorated Permutations:2751346 2751346 2751346 2751346Leveled permutrees:
1
2
34
5
6
7
7641 2 3 5 4
4
3
3
5
5
6
6
7
7
2
2
1
1 7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
Definition of a permutree
directed (bottom to top) and labeled (bijectively by [n]) tree suchthat
j
<j >j
?j
?
?
j<j >j
?
j<j >j
<j >j
1
2
34
5
6
7
7641 2 3 5 4
4
3
3
5
5
6
6
7
7
2
2
1
1 7
64
42
1
3
5
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
InsertionSn × { , , , }n −→ Permutrees
(σ, δ) −→ Pδ(σ)
Congruence. . . ac . . . b . . . ≡δ . . . ca . . . b . . . (δb ∈ { , }). . . b . . . ac . . . ≡δ . . . b . . . ca . . . (δb ∈ { , })
Property
σ ≡δ τ ⇔ Pδ(σ) = Pδ(τ)
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
InsertionSn × { , , , }n −→ Permutrees
(σ, δ) −→ Pδ(σ)
Congruence. . . ac . . . b . . . ≡δ . . . ca . . . b . . . (δb ∈ { , }). . . b . . . ac . . . ≡δ . . . b . . . ca . . . (δb ∈ { , })
Property
σ ≡δ τ ⇔ Pδ(σ) = Pδ(τ)
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
InsertionSn × { , , , }n −→ Permutrees
(σ, δ) −→ Pδ(σ)
Congruence. . . ac . . . b . . . ≡δ . . . ca . . . b . . . (δb ∈ { , }). . . b . . . ac . . . ≡δ . . . b . . . ca . . . (δb ∈ { , })
Property
σ ≡δ τ ⇔ Pδ(σ) = Pδ(τ)
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
7251346 ≡
1234567
7
64
42
1
3
5
2751346
≡ 2715346
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
7251346 ≡
1234567
7
64
42
1
3
5
2751346
≡ 2715346
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
1234567
7
64
42
1
3
5
7251346 ≡
1234567
7
64
42
1
3
5
2751346
≡ 2715346
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
1234567
7
64
42
1
3
5
7251346 ≡
1234567
7
64
42
1
3
5
2751346
≡ 2715346
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
DecorationsInsertionNumerology
1234567
7
64
42
1
3
5
7251346 ≡
1234567
7
64
42
1
3
5
2751346
1234567
7
64
42
1
35
≡ 2715346
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
Lattice definition
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
Lattice definition
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
4321
4231 43123421
34123241 2431 4213
1234
13242134 1243
31242314 13422143
3142 24133214 2341 4123
1423
4132
1432
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
The PermutreehedronThm (Pilaud, P.) for every δ ∈ { , , , }n, there is an explicitconstruction of a
I a complete simplicial fan, the δ-permutree fan F(δ);
I a polytope, the Permutreehedron PT(δ), whose normal fan isF(δ).
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
The PermutreehedronThm (Pilaud, P.) for every δ ∈ { , , , }n, there is an explicitconstruction of a
I a complete simplicial fan, the δ-permutree fan F(δ);
I a polytope, the Permutreehedron PT(δ), whose normal fan isF(δ).
The Permutreehedron can be con-structed by convex hull or hyperplaneintersection.
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
The PermutreehedronThm (Pilaud, P.) for every δ ∈ { , , , }n, there is an explicitconstruction of a
I a complete simplicial fan, the δ-permutree fan F(δ);
I a polytope, the Permutreehedron PT(δ), whose normal fan isF(δ).
The vertices of PT(δ) are the δ-permutrees.
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
The PermutreehedronThm (Pilaud, P.) for every δ ∈ { , , , }n, there is an explicitconstruction of a
I a complete simplicial fan, the δ-permutree fan F(δ);
I a polytope, the Permutreehedron PT(δ), whose normal fan isF(δ).
The oriented graph of PT(δ) is theHasse diagram of the δ-permutreelattice.
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
The PermutreehedronThm (Pilaud, P.) for every δ ∈ { , , , }n, there is an explicitconstruction of a
I a complete simplicial fan, the δ-permutree fan F(δ);
I a polytope, the Permutreehedron PT(δ), whose normal fan isF(δ).
Combinatorics of the faces: Schroderpermutrees.
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
The PermutreehedronThm (Pilaud, P.) for every δ ∈ { , , , }n, there is an explicitconstruction of a
I a complete simplicial fan, the δ-permutree fan F(δ);
I a polytope, the Permutreehedron PT(δ), whose normal fan isF(δ).
Show the 3D-surprise!
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
Matriochka Permutreehedrarefinement δ 4 δ′ =⇒ inclusion PT(δ) ⊂ PT(δ′)
Vincent Pilaud – Viviane Pons Permutrees
MotivationsDefinition and combinatorics
Lattice structureGeometry advertisement
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )
PT( )PT( )
PT( )
Vincent Pilaud – Viviane Pons Permutrees