enumeration of lattice walks

enumeration of lattice walks

Introduction

We present

explicit formulas

for the number of walks in certain lattices. Proofs are given in a separate entry.

Definitions

The following definitions formalize the concepts of square lattice, triangular

lattice

,

honeycomb

lattice, dice lattice, and walk.

Definition

The

square lattice

is the graph on

vertex set

Z

×

Z

with each vertex

(

i

,

j

)

adjacent

to vertices

(

i

+

1

,

j

)

,

(

i

-

1

,

j

)

,

(

i

,

j

+

1

)

and

(

i

,

j

-

1

)

.

Definition

The

triangular lattice

is the graph on vertex set

Z

×

Z

with each vertex

(

i

,

j

)

adjacent to vertices

(

i

+

1

,

j

)

,

(

i

-

1

,

j

)

,

(

i

,

j

+

1

)

,

(

i

,

j

-

1

,

(

i

-

1

,

j

+

1

)

, and

(

i

+

1

,

j

-

1

)

.

Definition

The

honeycomb lattice

is the graph on vertex set

{

(

i

,

j

)

∈

Z

×

Z

:

i

+

j

≢

2

(

mod

3

)

}

, and with the following adjacencies:

1.

If

i

+

j

≡

0

(

mod

3

)

, then

(

i

,

j

)

is adjacent to vertices

(

i

+

1

,

j

)

,

(

i

,

j

+

1

)

, and

(

i

-

1

,

j

-

1

)

.

2.

If

i

+

j

≡

1

(

mod

3

)

, then

(

i

,

j

)

is adjacent to vertices

(

i

-

1

,

j

)

,

(

i

,

j

-

1

)

, and

(

i

+

1

,

j

+

1

)

.

Definition

The

dice lattice

is the graph on vertex set

Z

×

Z

with the following adjacencies:

1.

If

i

+

j

≡

0

(

mod

3

)

, then

(

i

,

j

)

is adjacent to vertices

(

i

+

1

,

j

)

,

(

i

-

1

,

j

)

,

(

i

,

j

+

1

)

,

(

i

,

j

-

1

)

,

(

i

+

1

,

j

+

1

)

, and

(

i

-

1

,

j

-

1

)

.

2.

If

i

+

j

≡

1

(

mod

3

)

, then

(

i

,

j

)

is adjacent to vertices

(

i

-

1

,

j

)

,

(

i

,

j

-

1

)

, and

(

i

+

1

,

j

+

1

)

.

3.

If

i

+

j

≡

2

(

mod

3

)

, then

(

i

,

j

)

is adjacent to vertices

(

i

+

1

,

j

)

,

(

i

,

j

+

1

)

, and

(

i

-

1

,

j

-

1

)

.

Definition

A

walk

in a graph is a

sequence

of vertices where consecutive vertices in the sequence are adjacent. Notice that the same vertex can appear

multiple

times in a walk. A

closed walk

is a walk starting and ending with the same vertex. We use the notation

w

⁢

(

x

,

y

,

n

)

for the number of walks of length

n

from

(

0

,

0

)

to

(

x

,

y

)

in a lattice.

Square lattice

The number of walks

w

⁢

(

x

,

y

,

n

)

of length

n

from

(

0

,

0

)

to

(

x

,

y

)

∈

ℤ

×

ℤ

in the square lattice is given by

w

⁢

(

x

,

y

,

n

)

=

(

n

n

+

x

-

y

2

)

⁢

(

n

n

-

x

-

y

2

)

whenever

x

+

y

and

n

have the same parity. Otherwise

w

⁢

(

x

,

y

,

n

)

=

0

.

Triangular lattice

The number of walks

w

⁢

(

x

,

y

,

n

)

of length

n

from

(

0

,

0

)

to

(

x

,

y

)

∈

ℤ

×

ℤ

in the triangular lattice is given by

w

⁢

(

x

,

y

,

n

)

=

∑

k

=

0

n

(

-

2

)

n

-

k

⁢

(

n

k

)

⁢

∑

j

=

0

k

(

k

j

)

⁢

(

k

j

-

x

-

y

)

⁢

(

k

j

-

x

)

In particular, the number of closed walks of length

n

starting and ending at

(

0

,

0

)

is

w

⁢

(

0

,

0

,

n

)

=

∑

k

=

0

n

(

-

2

)

n

-

k

⁢

(

n

k

)

⁢

∑

j

=

0

k

(

k

j

)

3

Honeycomb lattice

The number of walks

w

⁢

(

x

,

y

,

n

)

of even length

n

from

(

0

,

0

)

to

(

x

,

y

)

∈

ℤ

×

ℤ

in the honeycomb lattice is given by

w

⁢

(

x

,

y

,

n

)

=

∑

k

=

0

n

/

2

(

n

/

2

k

)

⁢

∑

j

=

0

k

(

k

j

+

x

-

(

x

+

y

)

/

3

)

⁢

(

k

j

)

⁢

(

k

j

+

(

x

+

y

)

/

3

)

=

∑

k

=

0

n

/

2

(

n

/

2

k

)

⁢

(

n

/

2

k

+

(

x

+

y

)

/

3

)

⁢

(

2

⁢

k

+

(

x

+

y

)

/

3

k

+

(

2

⁢

x

-

y

)

/

3

)

whenever

x

+

y

≡

0

(

mod

3

)

. Otherwise w(x,y,n) = 0.

The number of walks of odd length

n

from

(

0

,

0

)

to

(

x

,

y

)

∈

ℤ

×

ℤ

is

w

⁢

(

x

,

y

,

n

)

=

w

⁢

(

x

-

1

,

y

,

n

-

1

)

+

w

⁢

(

x

,

y

+

1

,

n

-

1

)

+

w

⁢

(

x

+

1

,

y

+

1

,

n

-

1

)

if

x

+

y

≡

1

(

mod

3

)

. Otherwise

w

⁢

(

x

,

y

,

n

)

=

0

.

Setting

x

=

y

=

0

we obtain the following formula for the number of closed walks of (necessarily) even length

n

starting and ending at

(

0

,

0

)

:

w

⁢

(

0

,

0

,

n

)

=

∑

k

=

0

n

/

2

(

n

/

2

k

)

⁢

∑

j

=

0

k

(

k

j

)

3

=

∑

k

=

0

n

/

2

(

n

/

2

k

)

2

⁢

(

2

⁢

k

k

)

Dice lattice

The number of walks

w

⁢

(

x

,

y

,

n

)

of even length

n

from

(

0

,

0

)

to

(

x

,

y

)

in the dice lattice is given by

w

⁢

(

x

,

y

,

n

)

=

2

n

/

2

⁢

∑

k

=

0

n

/

2

(

n

/

2

k

)

⁢

∑

j

=

0

k

(

k

j

+

x

-

(

x

+

y

)

/

3

)

⁢

(

k

j

)

⁢

(

k

j

+

(

x

+

y

)

/

3

)

=

2

n

/

2

⁢

∑

k

=

0

n

/

2

(

n

/

2

k

)

⁢

(

n

/

2

k

+

(

x

+

y

)

/

3

)

⁢

(

2

⁢

k

+

(

x

+

y

)

/

3

k

+

(

2

⁢

x

-

y

)

/

3

)

whenever

x

+

y

≡

0

(

mod

3

)

. Otherwise

w

⁢

(

x

,

y

,

n

)

=

0

.

The number of walks of odd length

n

from

(

0

,

0

)

to

(

x

,

y

)

∈

ℤ

×

ℤ

is

w

⁢

(

x

,

y

,

n

)

=

w

⁢

(

x

+

1

,

y

,

n

-

1

)

+

w

⁢

(

x

,

y

+

1

,

n

-

1

)

+

w

⁢

(

x

-

1

,

y

-

1

,

n

-

1

)

if

x

+

y

≡

2

(

mod

3

)

; if

x

+

y

≡

1

(

mod

3

)

,

w

⁢

(

x

,

y

,

n

)

=

w

⁢

(

x

-

1

,

y

,

n

-

1

)

+

w

⁢

(

x

,

y

-

1

,

n

-

1

)

+

w

⁢

(

x

+

1

,

y

+

1

,

n

-

1

)

. Otherwise

w

⁢

(

x

,

y

,

n

)

=

0

.

Setting

x

=

y

=

0

, we obtain the following formula for the number of closed walks of (necessarily) even length

n

starting and ending at

(

0

,

0

)

:

w

⁢

(

0

,

0

,

n

)

=

2

n

/

2

⁢

∑

k

=

0

n

/

2

(

n

/

2

k

)

⁢

∑

j

=

0

k

(

k

j

)

3

=

2

n

/

2

⁢

∑

k

=

0

n

/

2

(

n

/

2

k

)

2

⁢

(

2

⁢

k

k

)

Title

enumeration of lattice walks

Canonical name

EnumerationOfLatticeWalks

Date of creation

2013-03-22 19:20:54

Last modified on

2013-03-22 19:20:54

Owner

csguy (26054)

Last modified by

csguy (26054)

Numerical id

5

Author

csguy (26054)

Entry type

Topic

Classification

msc 05A15

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