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1: 26.2 Basic Definitions

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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. …

2: 31.6 Path-Multiplicative Solutions

§31.6 Path-Multiplicative Solutions

►A further extension of the notation (31.4.1) and (31.4.3) is given by …These solutions are called path-multiplicative. …

3: 31.18 Methods of Computation

… ►Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …

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

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

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$m$	$n$
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5: 5.21 Methods of Computation

… ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …

6: 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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7: 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 $y=x$. …

8: 31.9 Orthogonality

… ►The integration path begins at $z=\zeta$, encircles $z=1$ once in the positive sense, followed by $z=0$ once in the positive sense, and so on, returning finally to $z=\zeta$. The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. … ►and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. … ►For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2). …

9: 9.17 Methods of Computation

… ►As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. … ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …

10: 26.9 Integer Partitions: Restricted Number and Part Size

… ►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.

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