# lattice paths

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## 9 matching pages

##### 1: 26.2 Basic Definitions
###### LatticePath
A lattice path is a directed path on the plane integer lattice $\{0,1,2,\ldots\}\times\{0,1,2,\ldots\}$. …For an example see Figure 26.9.2. A k-dimensional lattice path is a directed path composed of segments that connect vertices in $\{0,1,2,\dots\}^{k}$ so that each segment increases one coordinate by exactly one unit. …
##### 2: 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 $y=x$. …
##### 4: 26.3 Lattice Paths: Binomial Coefficients
###### §26.3(i) Definitions
$\genfrac{(}{)}{0.0pt}{}{m+n}{n}$ is the number of lattice paths from $(0,0)$ to $(m,n)$. …The number of lattice paths from $(0,0)$ to $(m,n)$, $m\leq n$, that stay on or above the line $y=x$ is $\genfrac{(}{)}{0.0pt}{}{m+n}{m}-\genfrac{(}{)}{0.0pt}{}{m+n}{m-1}.$
##### 5: 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 $(h,j)$, $1\leq h\leq m$, $1\leq j\leq k$, above and to the left of the lattice path. …
##### 6: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
###### §26.4(i) Definitions
It is also the number of $k$-dimensional lattice paths from $(0,0,\ldots,0)$ to $(n_{1},n_{2},\ldots,n_{k})$. …
##### 7: 23.6 Relations to Other Functions
where the integral is taken along any path from $z$ to $\infty$ that does not pass through any of $e_{1},e_{2},e_{3}$. Then $z=\wp\left(w\right)$, where the value of $w$ depends on the choice of path and determination of the square root; see McKean and Moll (1999, pp. 87–88 and §2.5). …
##### 8: 20.13 Physical Applications
20.13.1 $\ifrac{\partial\theta(z|\tau)}{\partial\tau}=\kappa\ifrac{{\partial}^{2}\theta% (z|\tau)}{{\partial z}^{2}},$
In the singular limit $\Im\tau\rightarrow 0+$, the functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
##### 9: Errata
• Equation (23.6.15)
23.6.15 $\frac{\sigma\left(u+\omega_{j}\right)}{\sigma\left(\omega_{j}\right)}=\exp% \left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)\frac{\theta_{j+1}% \left(z,q\right)}{\theta_{j+1}\left(0,q\right)},$ $j=1,2,3$

The factor $\exp\left(\eta_{j}u+\frac{\eta_{j}u^{2}}{2\omega_{1}}\right)$ has been corrected to be $\exp\left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)$.

Reported by Jan Felipe van Diejen on 2021-02-10

• Section 19.25(vi)

This subsection has been significantly updated. In particular, the following formulae have been corrected. Equation (19.25.35) has been replaced by

19.25.35 $z+2\omega=\pm R_{F}\left(\wp\left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp% \left(z\right)-e_{3}\right),$

in which the left-hand side $z$ has been replaced by $z+2\omega$ for some $2\omega\in\mathbb{L}$, and the right-hand side has been multiplied by $\pm 1$. Equation (19.25.37) has been replaced by

19.25.37 $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)=\pm 2R_{G}\left(\wp% \left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp\left(z\right)-e_{3}\right),$

in which the left-hand side $\zeta\left(z\right)+z\wp\left(z\right)$ has been replaced by $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)$ and the right-hand side has been multiplied by $\pm 1$. Equation (19.25.39) has been replaced by

19.25.39 $\zeta\left(\omega_{j}\right)+\omega_{j}e_{j}=2R_{G}\left(0,e_{j}-e_{k},e_{j}-e% _{\ell}\right),$

in which the left-hand side $\eta_{j}$ was replaced by $\zeta\left(\omega_{j}\right)$, for some $2\omega_{j}\in\mathbb{L}$ and $\wp\left(\omega_{j}\right)=e_{j}$. Equation (19.25.40) has been replaced by

19.25.40 $z+2\omega=\pm\sigma\left(z\right)R_{F}\left(\sigma_{1}^{2}(z),\sigma_{2}^{2}(z% ),\sigma_{3}^{2}(z)\right),$

in which the left-hand side $z$ has been replaced by $z+2\omega$, and the right-hand side was multiplied by $\pm 1$. For more details see §19.25(vi).

• Subsection 19.25(vi)

The Weierstrass lattice roots $e_{j},$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_{j}$, and lattice invariants $g_{2}$, $g_{3}$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

• Paragraph Mellin–Barnes Integrals (in §8.6(ii))

The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$. The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma\left(a,z\right)$. In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\ldots$.

Reported 2017-07-10 by Kurt Fischer.

• Equation (23.2.4)
23.2.4 $\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right)$

Originally the denominator $(z-w)^{2}$ was given incorrectly as $(z-w^{2})$.

Reported 2012-02-16 by James D. Walker.