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

(1)

Lattice congruences Permutrees

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

Permutrees

(2)

The Weak order The Permutree congruences

Weak Order

123

213 132

231 312

321

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413∧4213 = 2413

2413∨4213 = 4231

Vincent Pilaud – Viviane Pons Permutrees

(3)

Weak Order

123

213 132

231 312

321

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413∧4213 = 2413

2413∨4213 = 4231

(4)

Weak Order

123

213 132

231 312

321

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413∧4213 = 2413 2413∨4213 = 4231

(5)

Weak Order

123

213 132

231 312

321

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413∧4213 = 2413 2413∨4213 = 4231

(6)

Weak Order

123

213 132

231 312

321

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413∧4213 = 2413 2413∨4213 = 4231

(7)

Weak Order

123

213 132

231 312

321

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413∧4213 = 2413 2413∨4213 = 4231

(8)

Weak Order

123

213 132

231 312

321

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413∧4213 = 2413 2413∨4213 = 4231

(9)

Lattice congruence

The equivalence relation must be compatible with the lattice structure

x₁≡x₂ y1≡y2

⇒ (x₁∧y₁)≡(x₂∧y₂) (x1∨y1)≡(x2∨y2)

(10)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2134≡2314

2134∨2413≡2314∨2413 2431≡2413

(11)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2134≡2314

2134∨2413≡2314∨2413 2431≡2413

(12)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2134≡2314

2134∨2413≡2314∨2413

2431≡2413

(13)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2134≡2314

2134∨2413≡2314∨2413 2431≡2413

(14)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2134≡2314≡2341 2143≡2413≡2431 3214≡3241

4213≡4231

(15)

What are all the lattice congruence of the Weak order?

Lattice Congruences of the Weak Order, Nathan Reading,Order, 2005

(16)

the sylvester congruence and the Tamari lattice

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

(17)

the sylvester congruence and the Tamari lattice

1234 2134 1324 1243 2314 2143 1342

3214 2341 1432

3241 2431 3421

4321

(18)

The Permutree congruences

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

→ 24 classes

→ 18 classes

→ 12 classes

→ 18 classes

18 classes 12 classes

→ 14 classes

→ 8 classes

(19)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

→

24 classes

→ 18 classes

→ 12 classes

→ 18 classes

→ 14 classes

→ 8 classes

(20)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

→

24 classes

→

18 classes

→ 18 classes

→ 12 classes

→ 18 classes

→ 14 classes

→ 8 classes

(21)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

→

24 classes

→

18 classes

→

18 classes

→ 12 classes

→ 18 classes

→ 14 classes

→ 8 classes

(22)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

→

24 classes

→

18 classes

→

18 classes

→

12 classes

→ 18 classes

→ 14 classes

→ 8 classes

(23)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

→

24 classes

→

18 classes

→

18 classes

→

12 classes

→ 18 classes

→ 14 classes

→ 8 classes

(24)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

→

24 classes

→

18 classes

→

18 classes

→

12 classes

→

18 classes 18 classes 12 classes

→ 14 classes

→ 8 classes

(25)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

→

24 classes

→

18 classes

→

18 classes

→

12 classes

→

18 classes 18 classes 12 classes

→

14 classes

→ 8 classes

(26)

Numerology

8

10 10 10 10

12 14 14 14 14 12

18 18 18 18

24

(27)

4321

4231 4312

3421 3412

3241 2431 4213

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

(28)

Permutree insertion Permutree congruence Geometry

Permutrees

The permutrees are combinatorial objects which encode those lattice congruences.

(29)

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 1234 13242134 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 ?τ⁰

Fτ⊗Fτ0 4Fγ=P γcut

B(T, γ)⊗A(T, γ) 4Xη=P γcut

B(η, γ)⊗A(η, γ)

(30)

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 (Reading)

n ←→ binary sequences

(31)

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 (Reading)

n ←→ binary sequences

(32)

The Permutree insertion Permutation: 2751346 Decoration:

Decorated permutation: 2751346

(33)

(34)

12 34 56 7

7

6 4 4 2

1 3

5

(35)

12 34 56 7

7

6 4 4 2

1 3

5

(36)

12 34 56 7

7

6 4 4 2

1 3

5

(37)

12 34 56 7

7

6 4 4 2

1 3

5

(38)

12 34 56 7

7

6 4 4 2

1 3

5

(39)

3

12 34 56 7

7

6 4 4 2

1

5

(40)

12 34 56 7

7

6 4 4 2

1 3

5

(41)

12 34 56 7

7

6 4 4 2

1 3

5

(42)

7

6 4 4 2

1 3

5

(43)

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

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

(44)

The Permutree insertion Permutation : 2751346

Decorations:

Decorated Permutations:

2751346 2751346 2751346 2751346

Leveled permutrees:

(45)

Decorations:

2751346 2751346 2751346 2751346

Leveled permutrees:

1 2 3 4 5 6 7

1 2

3 4

5 6

7

(46)

Decorations:

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

(47)

Decorations:

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

(48)

Decorations:

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

(49)

Decorations:

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

(50)

Decorations:

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

(51)

Decorations:

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

(52)

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)

(53)

Insertion

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

Property

σ ≡_δ τ ⇔P_δ(σ) =P_δ(τ)

(54)

Insertion

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

Property

σ ≡_δ τ ⇔P_δ(σ) =P_δ(τ)

(55)

7251346≡

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

2751346

≡2715346

(56)

7251346≡

1 2 3 4 5 6 7

7

6 4 4 2

1 3

5

2751346

≡2715346

(57)

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

(58)

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

(59)

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

(60)

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

Decoration:

3214≡3241≡3421 2314≡2341

3124≡3142≡3412 1324≡1342

(61)

(62)

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).

(63)

The Permutreehedron

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

(64)

The Permutreehedron

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

(65)

The Permutreehedron

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

(66)

The Permutreehedron

Combinatorics of the faces: Schr¨oder permutrees.

(67)

Matriochka Permutreehedra

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

(68)

PT( )

PT( ) PT( )

PT( )

(69)

Main paper

V. Pilaud, V. Pons. Permutrees. Algebraic Combinatorics (2018).

Some background

N. Reading. Cambrian lattices. Advances in Mathematics (2006) G. Chˆatel, V. Pilaud, Cambiran Hopf algebras,Advances in Mathematics (2017)

New work

V. Pilaud, V. Pons, and D. Tamayo Jim´enez. Permutree sorting.

(2020) Next step Other types