CNRS&EcolePolytechnique–LRI,Univ.Paris-Sud VincentPilaud– VivianePons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Vincent Pilaud – Viviane Pons CNRS & Ecole Polytechnique – LRI, Univ. Paris-Sud

Permutrees

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Some background

I Reading, Cambiran Lattices (2006).

I Chatel-Pilaud,Cambiran Hopf algebra (2014).

I Pilaud-P., Permutrees (2016).

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

permutations binary trees binary sequences

Combinatorics

4321

4231 4312

3421 3412

3241 2431 4213 4132

1234

1324 1243

2134 2143

2314 3124 1342 1423

3142 2413 4123 1432

3214 2341

− − −

− + − + − −

− − +

− + + + − + + + −

+ + +

Geometry

3421 3412 4321 4312

2413 4213

3214 1432 1423

1342

1243 123413242134 2341

2431

23143124

− − −

− + −

+ − −

− − +

− + +

+ − +

+ + − + + +

Algebra

Malvenuto-Reutenauer algebra Loday-Ronco algebra Solomon algebra

FQSym = vecthFτ|τ∈Si PBT = vecthPT|T∈ BT i Rec = vecthXη|η∈ ±^∗i

Fτ·Fτ⁰=P

σ∈ττ¯⁰Fσ PT·PT⁰= P

T^%T⁰_≤T00≤T_- T⁰

PT⁰⁰ Xη·Xη⁰=Xη+η⁰+Xη−η0

4Fσ= P

σ∈τ ?τ0Fτ⊗Fτ⁰ 4Fγ=P

γcut

B(T, γ)⊗A(T, γ) 4Xη=P γcut

B(η, γ)⊗A(η, γ)

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

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)

Example

n ←→ permutations of [n]

n ←→ standard binary search trees

{ , }ⁿ ←→ Cambrian trees

n ←→ binary sequences

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

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) Example

n ←→ permutations of [n]

n ←→ standard binary search trees

{ , }ⁿ ←→ Cambrian trees

n ←→ binary sequences

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

12 34 56 7

7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

12 34 56 7

7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

12 34 56 7

7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

12 34 56 7

7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

12 34 56 7

7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

3

12 34 56 7

7

6 4 4 2

1

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

12 34 56 7

7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7 1 2 3 4 5 6 7

7 6 4 1 2 3 5

1 2 3 4 5 6 7

4 4

3 3

5 5

6 6

7 7

2 2

1 1

12 34 56 7

7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2

3 4

5 6

7

7 6 4

1 2 3 5 4

4

3 3

5 5

6 6

7 7

2 2

1

1 7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

Definition of a permutree

directed (bottom to top) and labeled (bijectively by [n]) tree such that

j

<j >j

? j

?

?

j

<j >j

?

j

<j >j

<j >j

1 2

3 4

5 6

7

7 6 4

1 2 3 5 4

4

3 3

5 5

6 6

7 7

2 2

1

1 7

6 4 4 2

1 3

5

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

Insertion

S_n× { , , , }ⁿ−→Permutrees (σ, δ)−→P_δ(σ)

Congruence

. . .ac. . .b. . .≡_δ . . .ca. . .b. . . (δb∈ { , }) . . .b. . .ac. . .≡_δ . . .b. . .ca. . . (δ_b∈ { , })

Property

σ ≡_δ τ ⇔P_δ(σ) =P_δ(τ)

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

Insertion

S_n× { , , , }ⁿ−→Permutrees (σ, δ)−→P_δ(σ) Congruence

. . .ac. . .b. . .≡_δ . . .ca. . .b. . . (δ_b∈ { , }) . . .b. . .ac. . .≡_δ . . .b. . .ca. . . (δ_b∈ { , })

Property

σ ≡_δ τ ⇔P_δ(σ) =P_δ(τ)

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

Insertion

S_n× { , , , }ⁿ−→Permutrees (σ, δ)−→P_δ(σ) Congruence

. . .ac. . .b. . .≡_δ . . .ca. . .b. . . (δ_b∈ { , }) . . .b. . .ac. . .≡_δ . . .b. . .ca. . . (δ_b∈ { , })

Property

σ ≡_δ τ ⇔P_δ(σ) =P_δ(τ)

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

7251346≡

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

2751346

≡2715346

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

7251346≡

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

2751346

≡2715346

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

7251346≡

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

2751346

≡2715346

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

7251346≡

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

2751346

≡2715346

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

7251346≡

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

2751346

1 2 3 4 5 6 7

7

6 4 4 2

1

3 5

≡2715346

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

Numerology

8

10 10 10 10

12 14 14 14 14 12

18 18 18 18

24

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Decorations Insertion Numerology

Factorial-Catalan numbers

C(δ) := number of permutrees with decorationδ.

I δ₁ andδ_ndo not affect the number of permutrees.

I Ifδ_i = ,C(δ) =C(δ[1. . .i])×C(δ[i. . .n]).

I Any can be changed into a without changingC(δ).

I Ifδ ∈ { , }ⁿ, then C(δ) is given by the recursive formula:

C(δ) = X

i∈δ⁻¹( )

C(δ\i)+ X

i∈δ⁻¹( )

C(δ[1. . .i−1])C(δ[i+ 1. . .i])

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Lattice definition Tree-angulation

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Lattice definition Tree-angulation

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432 4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432 4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432 4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

4321

4231 4312

3421 3412

3241 2431 4213

1234 1324

2134 1243

3124

2314 2143 1342

3142 2413

3214 2341 4123

1423 4132 1432

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Lattice definition Tree-angulation

Tree-angulation

7 4 6

2 1

3 5

7

6 5

4 4

3 2

1

8 0

7

6 5

4 4

3 2

1

8 0

vertices above/below/inside [0,8] ←→ decoration diangle enclosing j ←→ node labeledj trianglei <j <k with j below ←→ node labeledj trianglei <j <k with j above ←→ node labeledj quadrangle i <j⁻,j⁺<k ←→ node labeledj

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Lattice definition Tree-angulation

Tree-angulation flip

7

6 5

4 4

3 2

1

8 0

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Lattice definition Tree-angulation

Tree-angulation flip

7

6 5

4 4

3 2

1

8 0

7

6 5

4 4

3 2

1

8 0

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Lattice definition Tree-angulation

Tree-angulation flip

7

6 5

4 4

3 2

1

8 0

7

6 5

4 4

3 2

1

8 0

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Lattice definition Tree-angulation

Tree-angulation flip

7

6 5

4 4

3 2

1

8 0

7

6 5

4 4

3 2

1

8 0

7

6 5

4 4

3 2

1

8 0

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Lattice definition Tree-angulation

Tree-angulation flip

7

6 5

4 4

3 2

1

8 0

7

6 5

4 4

3 2

1

8 0

7

6 5

4 4

3 2

1

8 0

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

The Permutreehedron

Thm (Pilaud, P.)for everyδ ∈ { , , , }ⁿ, there is an explicit construction of a

I a complete simplicial fan, theδ-permutree fan F(δ);

I a polytope, the PermutreehedronPT(δ), whose normal fan is F(δ).

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

The Permutreehedron

Thm (Pilaud, P.)for everyδ ∈ { , , , }ⁿ, there is an explicit construction of a

I a complete simplicial fan, theδ-permutree fan F(δ);

I a polytope, the PermutreehedronPT(δ), whose normal fan is F(δ).

The Permutreehedron can be con- structed by convex hull or hyperplane intersection.

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

The Permutreehedron

Thm (Pilaud, P.)for everyδ ∈ { , , , }ⁿ, there is an explicit construction of a

I a complete simplicial fan, theδ-permutree fan F(δ);

I a polytope, the PermutreehedronPT(δ), whose normal fan is F(δ).

The vertices of PT(δ) are the δ- permutrees.

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

The Permutreehedron

Thm (Pilaud, P.)for everyδ ∈ { , , , }ⁿ, there is an explicit construction of a

I a complete simplicial fan, theδ-permutree fan F(δ);

I a polytope, the PermutreehedronPT(δ), whose normal fan is F(δ).

The oriented graph of PT(δ) is the Hasse diagram of the δ-permutree lattice.

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

The Permutreehedron

Thm (Pilaud, P.)for everyδ ∈ { , , , }ⁿ, there is an explicit construction of a

I a complete simplicial fan, theδ-permutree fan F(δ);

I a polytope, the PermutreehedronPT(δ), whose normal fan is F(δ).

Combinatorics of the faces: Schr¨oder permutrees.

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

The Permutreehedron

Thm (Pilaud, P.)for everyδ ∈ { , , , }ⁿ, there is an explicit construction of a

I a complete simplicial fan, theδ-permutree fan F(δ);

I a polytope, the PermutreehedronPT(δ), whose normal fan is F(δ).

Show the 3D-surprise!

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

Matriochka Permutreehedra

refinementδ4δ⁰ =⇒ inclusionPT(δ)⊂PT(δ⁰)

Vincent Pilaud – Viviane Pons Permutrees

Motivations Definition and combinatorics Lattice structure Geometry advertisement

PT( )

PT( )

PT( )

PT( )

PT( )

PT( )

PT( )

PT( )

PT( )

PT( )

PT( )

PT( )

PT( )

PT( ) PT( )

PT( )

Vincent Pilaud – Viviane Pons Permutrees