lattice path について

Words near each other

Dictionary Lists

mini英和辞書

翻訳と辞書　辞書検索 [ 開発暫定版 ]

スポンサードリンク

lattice path ：ウィキペディア英語版

lattice path

In combinatorics, a lattice path

L

\mathbb^d

of length

k

with steps in

S

is a sequence

v_0, v_1, \ldots, v_k \in \mathbb^d

such that each consecutive difference

v_i - v_

lies in

S

.
A lattice path may lie in any lattice in

\mathbb^d

,〔 but the integer lattice

\mathbb^d

is most commonly used.
An example of a lattice path in

\mathbb^2

of length 5 with steps in

S = \lbrace (2,0), (1,1), (0,-1) \rbrace

L = \lbrace (-1,-2), (0,-1), (2,-1), (2,-2), (2,-3), (4,-3) \rbrace

.
==North-East lattice paths==
A North-East (NE) lattice path is a lattice path in

\mathbb^2

with steps in

S = \lbrace (0,1), (1,0) \rbrace

. The

(0,1)

steps are called North steps and denoted by

N

's;
the

(1,0)

steps are called East steps and denoted by

E

's.
NE lattice paths most commonly begin at the origin. This convention allows us to encode all the information about a NE lattice path

L

in a single permutation word. The length of the word gives us the number of steps of the lattice path,

k

. The order of the

N

's and

E

's communicates the sequence of

L

. Furthermore, the number of

N

's and the number of

E

's in the word determines the end point of

L

.
If the permutation word for a NE lattice path contains

n

N

steps and

e

E

steps, and if the path begins at the origin, then the path necessarily ends at

(e,n)

. This follows because you have "walked" exactly

n

steps North and

e

steps East from

(0,0)

.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
■ウィキペディアで「lattice path」の詳細全文を読む

スポンサードリンク

翻訳と辞書 : 翻訳のためのインターネットリソース