Parking functions in types C and B

Parking functions in types C and B

Robin Sulzgruber joint work with Marko Thiel

Universit¨at Wien

Ellwangen, 23^rd March 2015

Outline

Regions of the Shi arrangement Finite torus

(parking functions)

Diagonally labelled paths Vertically labelled paths

Outline

Regions of the Shi arrangement

Finite torus

(parking functions)

Diagonally labelled paths Vertically labelled paths

Outline

Regions of the Shi arrangement Finite torus

(parking functions)

Diagonally labelled paths Vertically labelled paths

Outline

Regions of the Shi arrangement Finite torus

(parking functions)

Diagonally labelled paths

Vertically labelled paths

Outline

Regions of the Shi arrangement Finite torus

(parking functions)

Diagonally labelled paths Vertically labelled paths

Parking functions

Hyperplanes and reflections

Definition

Let V be a Euclidean vector space with inner product h., .i,α∈V a nonzero vector and k ∈Z.

Define an affine hyperplane

Hα,k ={x ∈V :hx, αi=k}. Lets_α,k be the reflection throughH_α,k.

sα,k(x) =x−2hx, αi −2k hα, αi α.

Parking functions

Hyperplanes and reflections

Definition

Let V be a Euclidean vector space with inner product h., .i,α∈V a nonzero vector and k ∈Z.

Define an affine hyperplane

Hα,k ={x ∈V :hx, αi=k}. Lets_α,k be the reflection throughH_α,k.

sα,k(x) =x−2hx, αi −2k hα, αi α.

Parking functions

Hyperplanes and reflections

Definition

Let V be a Euclidean vector space with inner product h., .i,α∈V a nonzero vector and k ∈Z.

Define an affine hyperplane

Hα,k ={x∈V :hx, αi=k}.

Lets_α,k be the reflection throughH_α,k.

sα,k(x) =x−2hx, αi −2k hα, αi α.

Parking functions

Hyperplanes and reflections

Definition

Let V be a Euclidean vector space with inner product h., .i,α∈V a nonzero vector and k ∈Z.

Define an affine hyperplane

Hα,k ={x∈V :hx, αi=k}.

Lets_α,k be thereflectionthrough H_α,k.

sα,k(x) =x−2hx, αi −2k hα, αi α.

Parking functions

Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α}for all α∈Φ, s_α,0(Φ)⊆Φ for allα ∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ. Let Φ⁺ denote the positive roots.

We only consider irreducible root systems.

Example

2e1

e₂+e₁ 2e2

e₂−e₁

The root system of typeC₂ in V =R².

Parking functions

Roots

Definition

Acrystallographic root systemof V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, s_α,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ. Let Φ⁺ denote thepositive roots. We only consider irreducible root systems.

Example

2e1

e₂+e₁ 2e2

e₂−e₁

The root system of typeC₂ in V =R².

Parking functions

Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ,

s_α,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ. Let Φ⁺ denote the positive roots. We only consider irreducible root systems.

Example

2e1

e₂+e₁ 2e2

e₂−e₁

The root system of typeC₂ in V =R².

Parking functions

Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, s_α,0(Φ)⊆Φ for allα∈Φ,

2hβ, αi

hα, αi ∈Z for all α, β∈Φ. Let Φ⁺ denote the positive roots. We only consider irreducible root systems.

Example

2e1

e₂+e₁ 2e2

e₂−e₁

The root system of typeC₂ in V =R².

Parking functions

Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, s_α,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ.

Let Φ⁺ denote the positive roots. We only consider irreducible root systems.

Example

2e1

e₂+e₁ 2e2

e₂−e₁

The root system of typeC₂ in V =R².

Parking functions

Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, s_α,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ.

Let Φ⁺ denote the positive roots. We only consider irreducible root systems.

Example

2e1

e₂+e₁ 2e2

e₂−e₁

Parking functions

Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, s_α,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ.

LetΦ⁺ denote the positive roots.

We only consider irreducible root systems.

Example

2e1

e₂+e₁ 2e2

e₂−e₁

The root system of typeC₂ in V =R².

Parking functions

Roots

Definition

A crystallographic root system of V is a finite spanning subset Φ⊆V of nonzero vectors such that

Rα∩Φ ={α,−α} for all α∈Φ, s_α,0(Φ)⊆Φ for allα∈Φ, 2hβ, αi

hα, αi ∈Z for all α, β∈Φ.

Let Φ⁺ denote the positive roots.

Example

2e1

e₂+e₁ 2e2

e₂−e₁

Parking functions

The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ⁺

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ⁺={e_j±ei : 1≤i <j ≤n}

∪ {2e_i : 1≤i ≤n}.

The coroot lattice is Qˇ =Zⁿ. The finite torus is

Zⁿ/(2n+ 1)Zⁿ.

Parking functions

The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding corootas

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ⁺

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ⁺={e_j±ei : 1≤i <j ≤n}

∪ {2e_i : 1≤i ≤n}.

The coroot lattice is Qˇ =Zⁿ. The finite torus is

Zⁿ/(2n+ 1)Zⁿ.

Parking functions

The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define thecoroot latticeas Qˇ = X

α∈Φ⁺

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ⁺={e_j±ei : 1≤i <j ≤n}

∪ {2e_i : 1≤i ≤n}.

The coroot lattice is Qˇ =Zⁿ. The finite torus is

Zⁿ/(2n+ 1)Zⁿ.

Parking functions

The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ⁺

ˇ αZ.

Define the finite torusas T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ⁺={e_j±ei : 1≤i <j ≤n}

∪ {2e_i : 1≤i ≤n}.

The coroot lattice is Qˇ =Zⁿ. The finite torus is

Zⁿ/(2n+ 1)Zⁿ.

Parking functions

The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ⁺

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ⁺={e_j±ei : 1≤i <j ≤n}

∪ {2e_i : 1≤i ≤n}.

The coroot lattice is Qˇ =Zⁿ. The finite torus is

Zⁿ/(2n+ 1)Zⁿ.

Parking functions

The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ⁺

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ⁺={e_j±ei : 1≤i <j ≤n}

∪ {2e_i : 1≤i ≤n}.

The coroot lattice is Qˇ =Zⁿ.

The finite torus is Zⁿ/(2n+ 1)Zⁿ.

Parking functions

The coroot lattice and the finite torus

Definition

For each root α∈Φ define the corresponding coroot as

ˇ

α= 2

hα, αiα.

Define the coroot lattice as Qˇ = X

α∈Φ⁺

ˇ αZ.

Define the finite torus as T = ˇQ/(h+ 1) ˇQ.

Example: Type C

The positive roots are given by Φ⁺={e_j±ei : 1≤i <j ≤n}

∪ {2e_i : 1≤i ≤n}.

The coroot lattice is Qˇ =Zⁿ. The finite torus is

Zⁿ/(2n+ 1)Zⁿ.

Parking functions

Classical parking functions

Definition

A classical parking function is an integer vector

f = (f₁,f₂, . . . ,f_n) with nonnegative entries such that there exists a permutation σ∈S_n with

f_σ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1)

σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

Parking functions

Classical parking functions

Definition

Aclassical parking function is an integer vector

f = (f₁,f₂, . . . ,f_n) with nonnegative entries such that there exists a permutation σ∈S_n with

f_σ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1)

σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

Parking functions

Classical parking functions

Definition

A classical parking function is an integer vector

f = (f₁,f₂, . . . ,f_n) with nonnegative entries such that there exists a permutation σ∈S_n with

f_σ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1)

σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

Parking functions

Classical parking functions

Definition

A classical parking function is an integer vector

f = (f₁,f₂, . . . ,f_n) with nonnegative entries such that there exists a permutation σ∈S_n with

f_σ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1) σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

Parking functions

Classical parking functions

Definition

A classical parking function is an integer vector

f = (f₁,f₂, . . . ,f_n) with nonnegative entries such that there exists a permutation σ∈S_n with

f_σ(i)≤i−1.

Example

f = (1,4,0,0,4,4,1) σ·f = (0,0,1,1,4,4,4)

≤(0,1,2,3,4,5,6)

Parking functions

Parking functions of type A and C

Proposition

The set of classical parking functions of length n is a natural system of representatives for the finite torus of type An−1.

Recall that Zⁿ/(2n+ 1)Zⁿ is the finite torus of typeC_n. Definition

We define parking functions of type C as integer vectors f = (f1,f2, . . . ,fn) where −n ≤f_i ≤n.

Parking functions

Parking functions of type A and C

Proposition

The set of classical parking functions of length n is a natural system of representatives for the finite torus of type An−1.

Recall that Zⁿ/(2n+ 1)Zⁿ is the finite torus of typeC_n. Definition

We define parking functions of type C as integer vectors f = (f1,f2, . . . ,fn) where −n ≤f_i ≤n.

Parking functions

Parking functions of type A and C

Proposition

The set of classical parking functions of length n is a natural system of representatives for the finite torus of type An−1.

Recall that Zⁿ/(2n+ 1)Zⁿ is the finite torus of typeC_n.

Definition

We define parking functions of type C as integer vectors f = (f1,f2, . . . ,fn) where −n ≤f_i ≤n.

Parking functions

Parking functions of type A and C

Proposition

The set of classical parking functions of length n is a natural system of representatives for the finite torus of type An−1.

Recall that Zⁿ/(2n+ 1)Zⁿ is the finite torus of typeC_n. Definition

We define parking functions of type C as integer vectors f = (f1,f2, . . . ,fn) where−n≤f_i ≤n.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_nand a permutation σ∈S_n such that

σi < σi+1

for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6. Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

Avertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6. Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli arise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6. Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6. Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i =1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3

The rises of π are i = 1,3,5,6.

Let σ=3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i =1,3,5,6.

Let σ=3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled Dyck paths

Definition

A vertically labelled Dyck path is a pair (π, σ) of a Dyck path π∈ D_n and a permutation σ∈S_n such that

σ_i < σ_i+1 for each rise i ofπ.

We calli a rise if thei-th North step is followed by a North step.

Example

3 4

1 7

2 5 6

The rises of π are i = 1,3,5,6.

Let σ= 3417256. The pair (π, σ) is a vertical labelling.

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,

0

,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,0,

0

,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (

1

,

4

,0,0,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,

4

,0,0,

4

,

4

,

1

)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,

4

,0,0,

4

,

4

,1)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,4,0,0,

4

,

4

,1)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,4,0,0,4,

4

,1)

Parking functions

From vertically labelled Dyck paths to parking functions

Vertically labelled Dyck paths

3 4

1 7

2 5 6 0

0 1 2 3 4 5 6

Classical parking functions There is a natural way to construct the parking function corresponding to a vertically labelled Dyck path.

f = (1,4,0,0,4,4,1)

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5. Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

Avertically labelled lattice path is a pair(π, σ)of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i = 2,3,4,5. Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5. Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i =2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4 2 3

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4 2 3 6

The rises of π are i =2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

The pair (π, σ) is a vertical labelling.

Parking functions

Vertically labelled lattice paths

Definition

A vertically labelled lattice path is a pair (π, σ) of a lattice path π∈ L_n from (0,0) to (n,n) and a signed permutationσ ∈H_n such that

σ_i < σ_i+1

for each risei of π, and such that 0< σ1

ifπ begins with a North step.

Example

1

−5

−4 2 3 6

The rises of π are i = 2,3,4,5.

Moreover, π begins with a North step.

Let σ= 1(−5)(−4)236. The pair (π, σ) is a vertical labelling.

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0

Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0

Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 Type C parking functions

There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (

0

,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,

4

,

4

,

−4

,

−4

,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,

4

,

4

,

−4

,−4,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,

4

,

4

,−4,−4,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,4,

4

,−4,−4,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,4,4,−4,−4,

4

)

Parking functions

From vertically labelled lattice paths to parking functions

Vertically labelled lattice paths

1

−5

−4 2 3 6

0 1 2 3 4 5 6 Type C parking functions There is a natural bijection between typeC parking functions and vertically labelled lattice paths.

f = (0,4,4,−4,−4,4)

Shi regions

The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e₁,2e₂,e₂+e₁,e₂−e₁}.

Shi regions

The Shi arrangement

Definition

We define theShi arrangementof the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e₁,2e₂,e₂+e₁,e₂−e₁}.

Shi regions

The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e ,2e ,e +e ,e −e }.

Shi regions

The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e₁,2e₂,e₂+e₁,e₂−e₁}.

Shi regions

The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e ,2e ,e +e ,e −e }.

Shi regions

The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e₁,2e₂,e₂+e₁,e₂−e₁}.

Shi regions

The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e ,2e ,e +e ,e −e }.

Shi regions

The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e₁,2e₂,e₂+e₁,e₂−e₁}.

Shi regions

The Shi arrangement

Definition

We define the Shi arrangement of the root system Φ as

Shi_Φ ={H_α,k :α∈Φ⁺,k = 0,1}.

The connected components of

V − [

H∈Shi_Φ

H

are called the regions of the Shi arrangement.

Example: ShiC2

In typeC2 we have

Φ⁺={2e ,2e ,e +e ,e −e }.