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##### 1: 26.13 Permutations: Cycle Notation
is $(1,3,2,5,7)(4)(6,8)$ in cycle notation. …In consequence, (26.13.2) can also be written as $(1,3,2,5,7)(6,8)$. … For the example (26.13.2), this decomposition is given by $(1,3,2,5,7)(6,8)=(1,3)(2,3)(2,5)(5,7)(6,8).$If $j, then $(j,k)$ is a product of $2k-2j-1$ adjacent transpositions: …Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $(1,3,2,5,7)(6,8)=(2,3)\*(1,2)\*(4,5)(3,4)(2,3)(3,4)(4,5)(6,7)(5,6)(7,8)\*(6,7)$: $\mathop{\mathrm{inv}}((1,3,2,5,7)(6,8))=11$.
##### 2: 3.7 Ordinary Differential Equations
where $\mathbf{A}(\tau,z)$ is the matrix
3.7.6 $\mathbf{A}(\tau,z)=\begin{bmatrix}A_{11}(\tau,z)&A_{12}(\tau,z)\\ A_{21}(\tau,z)&A_{22}(\tau,z)\end{bmatrix},$
and $\mathbf{b}(\tau,z)$ is the vector … Let $(a,b)$ be a finite or infinite interval and $q(x)$ be a real-valued continuous (or piecewise continuous) function on the closure of $(a,b)$. … If $q(x)$ is $C^{\infty}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. …
##### 3: 26.15 Permutations: Matrix Notation
If $(j,k)\in B$, then $\sigma(j)\neq k$. The number of derangements of $n$ is the number of permutations with forbidden positions $B=\{(1,1),(2,2),\ldots,(n,n)\}$. … For $(j,k)\in B$, $B\setminus[j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\ldots,n$. $B\setminus(j,k)$ denotes $B$ with the element $(j,k)$ removed. … Let $B=\{(j,j),(j,j+1)\>|\>1\leq j. …
##### 4: 14.27 Zeros
$P^{\mu}_{\nu}\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P^{\mu}_{\nu}\left(x\pm i0\right)$ has no zeros in the interval $(-\infty,-1)$. …
##### 5: 32.4 Isomonodromy Problems
32.4.4 $\mathbf{A}(z,\lambda)=(4\lambda^{4}+2w^{2}+z)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-i(4\lambda^{2}w+2w^{2}+z)\begin{bmatrix}0&-i\\ i&0\end{bmatrix}-\left(2\lambda w^{\prime}+\frac{1}{2\lambda}\right)\begin{% bmatrix}0&1\\ 1&0\end{bmatrix},$
32.4.5 $\mathbf{B}(z,\lambda)=\left(\lambda+\dfrac{w}{\lambda}\right)\begin{bmatrix}1&% 0\\ 0&-1\end{bmatrix}-\dfrac{iw}{\lambda}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}.$
32.4.6 $\mathbf{A}(z,\lambda)=-i(4\lambda^{2}+2w^{2}+z)\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}-2w^{\prime}\begin{bmatrix}0&-i\\ i&0\end{bmatrix}+\left(4\lambda w-\frac{\alpha}{\lambda}\right)\begin{bmatrix}% 0&1\\ 1&0\end{bmatrix},$
32.4.7 $\mathbf{B}(z,\lambda)=\begin{bmatrix}-i\lambda&w\\ w&i\lambda\end{bmatrix}.$
32.4.8 $\mathbf{A}(z,\lambda)=\begin{bmatrix}\tfrac{1}{4}z&0\\ 0&-\tfrac{1}{4}z\end{bmatrix}+\begin{bmatrix}-\tfrac{1}{2}\theta_{\infty}&u_{0% }\\ u_{1}&\tfrac{1}{2}\theta_{\infty}\end{bmatrix}\dfrac{1}{\lambda}+\begin{% bmatrix}v_{0}-\tfrac{1}{4}z&-v_{1}v_{0}\\ \ifrac{(v_{0}-\tfrac{1}{2}z)}{v_{1}}&\tfrac{1}{4}z-v_{0}\end{bmatrix}\frac{1}{% \lambda^{2}},$
##### 6: 26.6 Other Lattice Path Numbers
$D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … $M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. … $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … $r(n)$ is the number of paths from $(0,0)$ to $(n,n)$ that stay on or above the diagonal $y=x$ and are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. …
##### 7: 14.16 Zeros
where $m$, $n\in\mathbb{Z}$ and $\delta_{\mu}$, $\delta_{\nu}\in(0,1)$. … The zeros of $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ in the interval $(-1,1)$ interlace those of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$. … $P^{\mu}_{\nu}\left(x\right)$ has exactly one zero in the interval $(1,\infty)$ if either of the following sets of conditions holds: … For all other values of $\mu$ and $\nu$ (with $\nu\geq-\frac{1}{2}$) $P^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$. $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$ when $\nu>-1$, and at most one zero in the interval $(1,\infty)$ when $\nu<-1$.
##### 8: 4.37 Inverse Hyperbolic Functions
In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function $(t^{2}-1)^{1/2}$ assumes its principal value when $t\in(1,\infty)$. …
4.37.16 $\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z\right),$ $z/i\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$;
4.37.19 $\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right),$ $z\in\mathbb{C}\setminus(-\infty,1)$,
It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on $(-\infty,1]$. …
4.37.24 $\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;
##### 9: 24.1 Special Notation
 $j,k,\ell,m,n$ integers, nonnegative unless stated otherwise. … greatest common divisor of $m,n$. $k$ and $m$ relatively prime.
##### 10: Jim Pitman
Pitman has devoted much effort to promote the development of open access resources in the fields of probability and statistics. As a member of the Executive Committee of the Institute of Mathematical Statistics (IMS) from 2005 to 2008, he guided the IMS through implementation of a policy to promote open access to all of its professional journals, through systematic deposit of peer-reviewed final versions of all articles on arXiv. … He has published extensively on probability, stochastic processes, combinatorics and is a champion for open access to resources in mathematics. …