lattice%20paths

SearchAdvancedHelp

(0.002 seconds)

5 matching pages

1: 26.3 Lattice Paths: Binomial Coefficients

§26.3 Lattice Paths: Binomial Coefficients

►

§26.3(i) Definitions

… ► $\left(\genfrac{}{}{0.0pt}{}{m+n}{n}\right)$ is the number of lattice paths from $(0,0)$ to $(m,n)$. …The number of lattice paths from $(0,0)$ to $(m,n)$, $m\le n$, that stay on or above the line $y=x$ is $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)-\left(\genfrac{}{}{0.0pt}{}{m+n}{m-1}\right).$ … ►

Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

► ►►

$m$	$n$
	…

►

…

2: 26.6 Other Lattice Path Numbers

§26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

… ►

Motzkin Number $M(n)$

… ►

Narayana Number $N(n,k)$

… ►

Schröder Number $r(n)$

…

3: 26.2 Basic Definitions

… ►

Lattice Path

►A lattice path is a directed path on the plane integer lattice $\{0,1,2,\mathrm{\dots }\}\times \{0,1,2,\mathrm{\dots }\}$. …For an example see Figure 26.9.2. ►A k-dimensional lattice path is a directed path composed of segments that connect vertices in ${\{0,1,2,\mathrm{\dots }\}}^k$ so that each segment increases one coordinate by exactly one unit. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

► ►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
3	3	20	627	37	21637
…

►

4: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

►

§26.5(i) Definitions

… ►It counts the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$. … ►

Table 26.5.1: Catalan numbers.

► ►►►

$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
…
6	132	13	7 42900	20	65641 20420

►

…

5: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

Table 26.9.1: Partitions $p_k\left(n\right)$.

► ►►►

$n$	$k$
	…
8	0	1	5	10	15	18	20	21	22	22	22
…

►

… ►It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. … ►It is also equal to the number of lattice paths from $(0,0)$ to $(m,k)$ that have exactly $n$ vertices $(h,j)$, $1\le h\le m$, $1\le j\le k$, above and to the left of the lattice path. … ►

Figure 26.9.2: The partition $5+5+3+2$ represented as a lattice path.

…