### 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

e_{2}+e_{1}
2e2

e_{2}−e_{1}

The root system of typeC_{2} in
V =R^{2}.

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

e_{2}+e_{1}
2e2

e_{2}−e_{1}

The root system of typeC_{2} in
V =R^{2}.

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

e_{2}+e_{1}
2e2

e_{2}−e_{1}

The root system of typeC_{2} in
V =R^{2}.

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

e_{2}+e_{1}
2e2

e_{2}−e_{1}

The root system of typeC_{2} in
V =R^{2}.

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

e_{2}+e_{1}
2e2

e_{2}−e_{1}

The root system of typeC_{2} in
V =R^{2}.

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

e_{2}+e_{1}
2e2

e_{2}−e_{1}

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

e_{2}+e_{1}
2e2

e_{2}−e_{1}

The root system of typeC_{2} in
V =R^{2}.

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

e_{2}+e_{1}
2e2

e_{2}−e_{1}

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 =Z^{n}.
The finite torus is

Z^{n}/(2n+ 1)Z^{n}.

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 =Z^{n}.
The finite torus is

Z^{n}/(2n+ 1)Z^{n}.

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 =Z^{n}.
The finite torus is

Z^{n}/(2n+ 1)Z^{n}.

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 =Z^{n}.
The finite torus is

Z^{n}/(2n+ 1)Z^{n}.

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 =Z^{n}.
The finite torus is

Z^{n}/(2n+ 1)Z^{n}.

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 =Z^{n}.

The finite torus is
Z^{n}/(2n+ 1)Z^{n}.

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 =Z^{n}.
The finite torus is

Z^{n}/(2n+ 1)Z^{n}.

Parking functions

### Classical parking functions

Definition

A classical parking function is an integer vector

f = (f_{1},f_{2}, . . . ,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 = (f_{1},f_{2}, . . . ,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 = (f_{1},f_{2}, . . . ,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

_{1},f_{2}, . . . ,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

_{1},f_{2}, . . . ,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 Z^{n}/(2n+ 1)Z^{n} 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 Z^{n}/(2n+ 1)Z^{n} 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 Z^{n}/(2n+ 1)Z^{n} 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

Recall that Z^{n}/(2n+ 1)Z^{n} 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_{n}and 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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

_{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

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

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

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

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^{+}={2e_{1},2e_{2},e_{2}+e_{1},e_{2}−e_{1}}.

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^{+}={2e_{1},2e_{2},e_{2}+e_{1},e_{2}−e_{1}}.

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^{+}={2e_{1},2e_{2},e_{2}+e_{1},e_{2}−e_{1}}.

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^{+}={2e_{1},2e_{2},e_{2}+e_{1},e_{2}−e_{1}}.

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^{+}={2e_{1},2e_{2},e_{2}+e_{1},e_{2}−e_{1}}.

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 }.