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