relation%20to%20lattice%20paths

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1: 26.6 Other Lattice Path Numbers

§26.6 Other Lattice Path Numbers

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Delannoy Number $D(m,n)$

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Motzkin Number $M(n)$

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Narayana Number $N(n,k)$

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Schröder Number $r(n)$

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2: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

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

§26.5(iii) Recurrence Relations

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26.5.6 $$C\left(n\right)\sim \frac{4^n}{\sqrt{\pi n^3}},$$ $n\to \mathrm{\infty }$,

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Symbols:

$C\left(n\right)$: Catalan number, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iv), §26.5 and Ch.26

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3: 26.3 Lattice Paths: Binomial Coefficients

§26.3 Lattice Paths: Binomial Coefficients

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§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 $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)-\left(\genfrac{}{}{0.0pt}{}{m+n}{m-1}\right).$ … ►

§26.3(iii) Recurrence Relations

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4: 26.9 Integer Partitions: Restricted Number and Part Size

… ►Conjugation establishes a one-to-one correspondence between partitions of $n$ into at most $k$ parts and partitions of $n$ into parts with largest part less than or equal to $k$. 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. … ►

§26.9(iii) Recurrence Relations

… ►As $n\to \mathrm{\infty }$ with $k$ fixed, …