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\title{Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises}

\author[1]{Jean-Luc Baril}
\author[2]{Jos{\'e} L. Ram{\'\i}rez}
\affil[1]{\rm LIB, Universit\'e de Bourgogne \protect\\
  B.P. 47 870, 21078 Dijon Cedex France\protect\\
   {\tt E-mail: barjl@u-bourgogne.fr
   }
}

\affil[2]{\rm Departamento de Matem{\'a}ticas, Universidad  Nacional de Colombia\protect\\
 Bogot{\'a}, Colombia\protect\\
   {\tt E-mail: jlramirezr@unal.edu.co
   }
}
\date{\today}
\providecommand{\keywords}[1]{\textit{Keywords:} #1}
\providecommand{\subjclass}[1]{\textit{MSC:} #1}

\date{\today}
\begin{document}

\maketitle

\begin{abstract}
 Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in $\mathbb{N}^2$ starting at the origin made of steps $U=(1,1)$, $D_k=(1,-k)$, $k\geq 1$ and $H=(1,0)$, where two down-steps cannot be consecutive. We enumerate MAP and their prefixes  avoiding peaks (resp. valleys, resp. double rise)  according to the length, the type of the last step, and the height of its end-point.  We express our results using Riordan arrays. Finally, we provide constructive bijections between these paths and restricted Dyck and Motzkin paths.
\end{abstract}

\section{Introduction}
In a recent  paper \cite{bakimava}, the authors introduce, study, and enumerate special classes of lattice paths, called {\it Dyck paths with air pockets} (DAP for short). Such paths are  non empty lattice paths in the first quadrant of $\mathbb{Z}^2$ starting at the origin, and consisting of up-steps $U=(1,1)$ and down-steps $D_k=(1,-k)$, $k\geq 1$, where two down-steps cannot be consecutive. These paths can be viewed as ordinary Dyck paths (i.e., paths in $\mathbb{N}^2$ starting at the origin, ending on the $x$-axis and consisting of $U$ and $D_1(=D)$), where each maximal run of down-steps is condensed into one large down-step. As mentioned in \cite{bakimava}, they also correspond to a stack evolution with (partial) reset operations that cannot be consecutive (see for instance \cite{krin}). The authors enumerate these paths with respect to the length, the type (up or down) of the last step and the height of the end-point. Whenever the last point is on the $x$-axis, they prove that the DAP of length $n$ are in one-to-one correspondence with the peak-less Motzkin paths of length $n-1$.  They also investigate the popularity of many patterns in these paths and they give asymptotic approximations. In a second work \cite{bkmv}, the authors make a study for a generalization of these paths by allowing them to go below the $x$-axis. They call these paths Grand Dyck paths with air pockets (GDAP), and they also yield enumerative results for these paths according to the length and several restrictions on the height. In a third paper, Baril and Barry \cite{BaBa} study two generalizations of DAP by allowing some horizontal steps $H=(1,0)$ with some conditions. They call them {\it Motzkin paths with air pockets of the first and second kind}.

In this paper we study Motzkin paths with air pockets of the first kind, which are defined as Motzkin paths (lattice paths in $\mathbb{N}^2$ starting at the origin and made of $U$, $D$, and $H$), where each maximal run of down-steps is condensed into one large down-step. More precisely, we consider lattice paths in $\mathbb{N}^2$ starting at the origin, consisting of steps $U$, $H$, and $D_k$, $k\geq 1$, where {\it two down-steps cannot be consecutive}.
We denote by $\mathcal{D}$ (resp. $\mathcal{M}$, resp. $\mathcal{MP}$) the set of Dyck paths (resp. Motzkin paths, resp. Motzkin paths with air pockets of the first kind).  Moreover, we denote by $\mathcal{PMP}$ the set of partial Motzkin paths with air pockets (PMAP for short). The MAP of the first kind are enumerated by the sequence \seqnum{A114465}. This sequence also counts the Dyck paths  having no ascents of length 2 that start at an odd level.

Throughout the paper,  we will use the following notations. For $k\geq 0$, we consider the generating function $f_k=f_k(z)$ (resp. $g_k=g_k(z)$, resp. $h_k=h_k(z)$), where the coefficient of $z^n$ in the series expansion is the number of  partial Motzkin paths with air pockets of length $n$  ending at height $k$ with an up-step, (resp. with a down-step,   resp. with a horizontal step $H$).
We introduce the bivariate generating functions
$$F(u,z)=\sum\limits_{k\geq 0} u^kf_k(z), \quad G(u,z)=\sum\limits_{k\geq 0} u^kg_k(z), \mbox{ and } H(u,z)=\sum\limits_{k\geq 0} u^kh_k(z).$$
For short, we also use the notation $F(u), G(u)$, and $H(u)$ for these functions.

A {\em Riordan array} is an infinite lower triangular matrix whose $k$-th column has generating function
$g(z)f(z)^k$ for all $k \ge 0$, for some formal power series  $g(z)$ and $f(z)$, with $g(0) \neq 0$, $f(0)=0$, and $f'(0)\neq 0$. Such a Riordan array is denoted by  $(g(z),f(z))$. We refer to \cite{Riordan, Riordan2} for more details on  Riordan arrays.  Several authors have used Riordan arrays to study lattice paths; see for example \cite{Sprugnoli2,   RAM, RAM2,  Yang, Yang2,  Yang3}.

The outline of this paper is the following. We present enumerative results for partial Motzkin paths with air pockets of the first kind avoiding peaks (resp. avoiding valleys, resp. avoiding double rises), knowing that a {\it peak} is an occurrence $UD_k$ for some $k\geq 1$, a {\it valley} is an occurrence $D_kU$ for some $k\geq 1$, and a {\it double rise} is an occurrence $UU$. For each avoidance, we provide bivariate generating functions that count the PMAP with respect to the length, the type of the last step (up, down or horizontal step) and the height of the end-point. All these results are obtained algebraically by using the famous kernel method for solving several systems of functional equations. We express our results using Riordan arrays and we deduce closed forms for  PMAP of length $n$ ending at height $k$. Finally, we provide constructive bijections between these paths and some restricted Dyck and Motzkin paths.

\section{Partial peak-less Motzkin paths with air pockets}

In this section, we study partial Motzkin paths with air pockets of the first kind avoiding occurrences of $UD_i$ for all $i\geq 1$.
\subsection{Enumerative results}
 Let $P$ be a length $n$ PMAP ending at height $k\geq 0$ and avoiding the occurrences of $UD_i$ for $i\geq 1$.
 If the last step of $P$ is $U$, then $k\geq 1$ and we have $P=QU$, where $Q$ is a length $(n-1)$ MAP ending at height $k-1$ and avoiding the peaks ($Q$ can be the empty path). So, we obtain the first relation  $f_k=zf_{k-1}+zg_{k-1}+zh_{k-1}$ for $k\geq 1$, anchored with $f_0=1$ by considering the empty path.  If the last step of $P$ is a down-step $D_i$, $i\geq 1$, then we have $P=QD_i$, where $Q$ is a length $(n-1)$ PMAP ending at height $\ell\geq k+1$ with no up- and down-steps at its end, and with no peaks. So, we obtain the second relation  $g_k=z\sum\limits_{\ell\geq k+1} h_\ell$. If the last step of $P$ is a horizontal step $H$,  then we have $P=QH$, where $Q$ is a length $(n-1)$ PMAP ending at height $k$ with no peaks, which implies that $h_k=z(f_k+g_k+h_k)$ for $k\geq 0$.

 Therefore, we have to solve the following system of equations:

\begin{equation}\left\{\begin{array}{l}
f_0=1,\mbox{ and } f_k=zf_{k-1}+zg_{k-1}+zh_{k-1}, \quad k\geq 1,\\
g_k=z\sum\limits_{\ell\geq k+1} h_\ell,\quad k\geq 0,\\
h_k=zf_k+zg_k+zh_k,\quad k\geq 0.\\
\end{array}\right.\label{equ1a}
\end{equation}

Multiplying by $u^k$ the recursions in (\ref{equ1a}) and summing over $k$, we have:
\begin{align*}
F(u)&=1+z\sum\limits_{k\geq 1}u^kf_{k-1}+z\sum\limits_{k\geq 1}u^kg_{k-1} +z\sum\limits_{k\geq 1}u^kh_{k-1}\\
&=1+zuF(u)+zuG(u)+zuH(u),\\
G(u)&=z\sum\limits_{k\geq 0}u^k \Bigl(\sum\limits_{\ell\geq k+1} h_\ell     \Bigr)=z\sum\limits_{k\geq 1}h_k(1+u+\cdots + u^{k-1})\\
&=z\sum\limits_{k\geq 1}\frac{u^k-1}{u-1}h_k=\frac{z}{u-1}(H(u)-H(1)),\\
H(u)&=zF(u)+zG(u)+zH(u).
\end{align*}

Notice that we have $F(1)-H(1)=1$ by considering the difference of the first and third equations. Now, setting $h1:=H(1)$ and solving these functional equations, we obtain

$$F(u)={\frac {{\it h1}\,u{z}^{2}+zu+{z}^{2}-u-z+1}{{u}^{2}z+{z}^{2}-u-z+1}},$$
$$G(u)=-{\frac {z \left( {\it h1}\,uz+z{\it h1}-{\it h1}+z \right) }{{u}^{2}z
+{z}^{2}-u-z+1}},\quad
H(u)={\frac {z \left( z{\it h1}-u+1 \right) }{{u}^{2}z+{z}^{2}-u-z+1}}.$$

In order to compute $h1$, we use the kernel method (see~\cite{ban, pro}) on $H(u)$. We can write the denominator (which is a polynomial in $u$ of degree 2), as $z(u-r)(u-s)$ with

$$r={\frac {1+\sqrt {-4\,{z}^{3}+4\,{z}^{2}-4\,z+1}}{2z}} \mbox{ and }
s={\frac {1-\sqrt {-4\,{z}^{3}+4\,{z}^{2}-4\,z+1}}{2z}}.$$
Plugging $u=s$ (which has a Taylor expansion at $z=0$) in $H(u)z(u-r)(u-s)$, we obtain the equation
$zh1-s+1 =0,$ which implies that $$h1=\frac{s-1}{z}.$$
Finally,  after simplifying by the factor $(u-s)$ in the numerators and denominators, we obtain
 $$F(u)=\frac{r}{r-u},\quad G(u)=\frac{s-1}{r-u},\quad\mbox{ and } \quad H(u)=\frac{1}{r-u},$$
which induces that
$$f_k=[u^k] F(u)=\frac{1}{r^k}, \quad g_k=[u^k] G(u)=\frac{s-1}{r^{k+1}},\quad \mbox{ and }\quad
h_k=[u^k] H(u)=\frac{1}{r^{k+1}}.
$$

\begin{thm} The bivariate generating function for the total number of peak-less PMAP with respect to the  length  and the height of the end-point is given by
$$\mathit{Total}(z,u)=\frac{1}{z(r-u)},$$ and we have
$$[u^k] \mathit{Total}(z,u)=\frac{1}{zr^{k+1}}.$$
Finally, setting $t(n,k)=[z^n][u^k]\mathit{Total}(z,u)$, we have for $n\geq 2$ and $k\geq 1$,  $$t(n,k)=t(n,k-1)+t(n-1,k)-t(n-1,k-2)-t(n-2,k),
$$
and setting $t_n:=t(n,0)$, then we have $$t_n=t_{n-1}+\sum\limits_{k=0}^{n-3}t_{k}t_{n-k-3}+\sum\limits_{k=2}^{n-1}\left(t_k-t_{k-1}\right)t_{n-k-1}.$$
\end{thm}
\noindent {\it Proof.} The first three equalities are immediately deduced from the previous results. Now, let us prove the last equality. Any length $n$ peak-less MAP is of the form ($i$) $HP$ where $P$ is a MAP of length $n-1$, or ($ii$) $U Q HD R$, where $Q,R$ are some MAP such that the length of $Q$ lies into $[0,n-3]$, or ($iii$) $P^\sharp Q$,  where $P^\sharp=UP'D_{i}$, $i\geq 2$, and $P'D_{i-1}$ is a MAP  of length lying into $[2,n-1]$. The number of $P'D_{i-1}$ of a given length $k$ is the total number of peak-less MAP of length $k$ minus the number of peak-less MAP of length $k$ and ending with $H$. Taking into account all these cases, we obtain the result.
\hfill $\Box$

\begin{cor} The generating function that counts all peak-less PMAP  with respect to the length is given by
$$\mathit{Total}(z,1)=\frac{1}{z(r-1)}.$$
\end{cor}

The first few terms of the series expansion of $\mathit{Total}(z,1)$ are
$$1 + 2z + 4z^2 + 9z^3 + 22z^4 + 56z^5 + 146z^6 + 388z^7 + 1048z^8 + 2869z^9 + O(x^{10}),$$
which corresponds to the sequence \oeis{A152225} in \cite{oeis} counting  Dyck paths of semilength $n+1$ with no peaks of height $0$ (mod 3) and no valleys of height 2 (mod 3); see \cite{Liu}.

\begin{cor} The generating function that counts the peak-less MAP with respect to the length is given by
$$\mathit{Total}(z,0)=\frac{1}{zr}.$$
\label{cor2}
\end{cor}
The first few terms of the series expansion of $\mathit{Total}(z,0)$ are
$$1 + z + z^2 + 2z^3 + 5z^4 + 12z^5 + 29z^6 + 73z^7 + 190z^8 + 505z^9+ O(z^{10}),$$ which corresponds to the sequence \oeis{A152171} in \cite{oeis} counting Dyck paths of length $2n$ with no peaks of height $2$ (mod 3) and no valleys of height 1 (mod 3). In Section 2.2, we will exhibit a constructive bijection between these two classes of paths.

\medskip

Let $\mathcal{T}$ be the infinite matrix $\mathcal{T}:=[t(n,k)]_{n, k\geq 0}$, where $t(n,k)=[z^n][u^k]\mathit{Total}(z,u)$. The first few rows of the matrix $\mathcal{T}$ are
\[\mathcal{T}=
\left(
\begin{array}{ccccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
 2 & 3 & 3 & 1 & 0 & 0 & 0 & 0 & 0 \\
 5 & 6 & 6 & 4 & 1 & 0 & 0 & 0 & 0 \\
 12 & 15 & 13 & 10 & 5 & 1 & 0 & 0 & 0 \\
 29 & 38 & 33 & 24 & 15 & 6 & 1 & 0 & 0 \\
 73 & 96 & 87 & 63 & 40 & 21 & 7 & 1 & 0 \\
 190 & 248 & 229 & 172 & 110 & 62 & 28 & 8 & 1 \\
\end{array}
\right).\]

In Figure \ref{fig1} we show the peak-less PMAP counted by $t(5,1)=15$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.85]{Ex1.eps}
\caption{Peak-less PMAP of length $5$ ending at height $1$.} \label{fig1}
\end{figure}

\begin{cor}
The matrix $\mathcal{T}=[t(n,k)]_{n, k\geq 0}$ is a Riordan array defined by
$$\left(C\left(z(1-z+z^2)\right),zC\left(z(1-z+z^2)\right)\right),$$
where $C(z)=\frac{1-\sqrt{1-4z}}{2z}$ is the generating function of the Catalan numbers
$c_n=\frac{1}{n+1}\binom{2n}{n}$.
\end{cor}
\noindent {\it Proof.}
 Indeed, we directly deduce the result from the following.
\begin{align*}
[u^k]\mathit{Total}(z,u)&=\frac{1}{zr^{k+1}}=\frac{1}{zr}\cdot \frac{1}{r^k}=C\left(z(1-z+z^2)\right)\cdot\left( zC\left(z(1-z+z^2)\right)\right)^k .
\end{align*}
\hfill $\Box$

\begin{cor}\label{corformula}
We have
$$t(n,k)=\sum_{j=0}^{n-k}\frac{k+1}{2(n-j)-k+1}\binom{2(n-j)-k+1}{n-k-j}a(n-k-j,j),$$
where $a(n,k)=(-1)^k\sum_{i=0}^n\binom ni \binom{n - i}{k - 2 i}$.
\end{cor}
\noindent {\it Proof.}
From the Lagrange Inversion Formula (cf. \cite{Lagrange}), we know that
\begin{align}\label{powCat}
C(z)^k=\sum_{n\geq 0}\frac{k}{2 n + k}\binom{2 n + k}{n}z^n.
\end{align}
From definition of Riordan arrays and Eq. \eqref{powCat} we have
\begin{align*}
t(n,k)&=[z^{n-k}]\left(C\left(z(1-z+z^2)\right)\right)^{k+1}\\
&=[z^{n-k}]\sum_{\ell=0}^{\infty}\frac{k+1}{2\ell+k+1}\binom{2\ell+k+1}{\ell}z^\ell(1-z+z^2)^\ell\\
&=[z^{n-k}]\sum_{\ell=0}^{\infty}\frac{k+1}{2\ell+k+1}\binom{2\ell+k+1}{\ell}z^\ell\sum_{j=0}^{2\ell}a(\ell,j)z^j\\
&=[z^{n-k}]\sum_{j=0}^{\infty}\sum_{\ell=0}^{\infty}\frac{k+1}{2(\ell+\lfloor j/2\rfloor)+k+1}\binom{2(\ell+\lfloor j/2\rfloor)+k+1}{\ell+\lfloor j/2\rfloor}a(\ell+\lfloor j/2\rfloor,j)z^{j+\ell+\lfloor j/2\rfloor}. \end{align*}
If we take $s=j+\ell+\lfloor j/2\rfloor$, then
\begin{align*}
t(n,k)&=[z^{n-k}]\sum_{j=0}^{\infty}\sum_{s=j}^{\infty}\frac{k+1}{2(s-j)+k+1}\binom{2(s-j)+k+1}{s-j}a(s-j,j)z^{s}\\
&=\sum_{j=0}^{n-k}\frac{k+1}{2(n-j)-k+1}\binom{2(n-j)-k+1}{n-k-j}a(n-k-j,j). \end{align*}
\hfill $\Box$

In \cite{Rogers}, Rogers gave an equivalent  characterization of the Riordan arrays. That is, every element not belonging to row $0$ or column $0$ in a Riordan array  can be expressed as a fixed linear combination of the elements in the preceding row. The \emph{$A$-sequence} is defined to be the sequence coefficients of this linear combination. Analogously, Merlini et al. \cite{Merlini}  introduced the \emph{$Z$-sequence}, that characterizes the elements in column $0$, except for the top one.

An infinite lower triangular matrix $[d_{n,k}]_{n,k\geq0}$ is a Riordan array if and only if  $d_{0,0}\neq 0$ and there exist
 two sequences $(a_0, a_1, a_2, \dots)$, with $a_0\neq 0$, and  $(z_0, z_1, z_2, \dots)$ (called the $A$-sequence and the $Z$-sequence, respectively), such that
\begin{align}\label{AZseq}
d_{n+1,k+1}&=a_0d_{n,k} + a_1d_{n,k+1} + a_2d_{n,k+2} +\cdots & \text{for } n, k\ge0,  \\
d_{n+1,0}&=z_0d_{n,0} + z_1d_{n,1} + z_2d_{n,2} +\cdots & \text{for } n\ge0.
\end{align}
The product of two Riordan arrays $(g(z),f(z))$ and $(h(z),l(z))$ is defined by
\begin{equation}\label{eq:product} (g(z),f(z))*(h(z),l(z))=\left(g(z)h(f(z)), l(f(z))\right). \end{equation}
Under this operation, the set of all Riordan arrays is a group  \cite{Riordan}. The identity element is $I=(1,z)$ and the inverse of $(g(z),f(z))$ is given by
 \begin{equation}\label{invRiordan}
 (g(z), f(z))^{-1}=\left(1/\left(g\circ f^{<-1>}\right)(z), f^{<-1>}(z)\right),
\end{equation}
  where $f^{<-1>}(z)$ denotes the compositional inverse of $f(z)$.

The generating functions for the $A$-sequence and $Z$-sequence of the Riordan array $\mathcal{F}=(g(z),f(z))$,  with inverse $\mathcal{F}^{-1}=(d(z),h(z))$, are given by  (\cite{Merlini, HeS})
\begin{align*}
A(z)=\frac{z}{h(z)} \quad \text{and} \quad Z(z)=\frac{1}{h(z)}\left(1-d_{0,0}d(z)\right),
\end{align*}
respectively.

From the definition of the $A$-sequence and $Z$-sequence for the Riordan arrays we can give an additional recurrence relation for the sequence $t(n,k)$.
\begin{cor}\label{corformula2}
We have
$$t(n+1,k+1)=\sum_{j\geq 0}a(j)t(n,k+j),$$
where $a(n)=(-1)^{n + 1}\sum_{k=1}^n \sum_{j=0}^k \frac{1}{k}\binom{j}{n - k -j}\binom{k}{j}\binom{n - k - 2}{k -1}$ for $n\geq 1$ and $a(0)=1$. Moreover,
$$t_{n+1}=\sum_{j\geq 0} a(j+1)t(n,j).$$
\end{cor}
\noindent {\it Proof.}
By Equation~\eqref{invRiordan}, the inverse of the matrix $ \mathcal{T}=[t(n,k)]_{n, k\geq 0}$ is given by
$\mathcal{T}^{-1}=\left(g_2(z), zg_2(z)\right)$,
where
$$g_2(z)=\frac{-1 + z^2 + \sqrt{1 - 2 z^2 + 4 z^3 - 3 z^4}}{2z^3}.$$
Therefore, the $A$-sequence and $Z$-sequence of the Riordan array $\mathcal{T}$ have generating functions
\begin{align*}
A(z)=\sum_{n\geq 0}a(n)z^n=\frac{2 z^3}{-1 + z^2 + \sqrt{1 - 2 z^2 + 4 z^3 - 3 z^4}} \quad \text{ and } \quad
Z(z)=\frac{A(z)-1}{z}.
\end{align*}
The generating function $A(z)$ corresponds with the sequence \seqnum{A247162}, where the explicit formula for $a(n)$ can be found.  From \eqref{AZseq} we obtain the result.
\hfill $\Box$

The first few values of the sequence $a(n)$ for $n\ge0$ are
$$1, \quad 1, \quad 0, \quad 1, \quad 0,\quad  1, \quad -1, \quad 2, \quad -3, \quad 6, \quad -10, \dots.$$

\subsection{A bijective approach}

Corollary~\ref{cor2} proves that the set of peak-less Motzkin paths with air pockets of length $n$ (ending on the $x$-axis) is equinumerous to the set $\mathcal{D}_n(2,1)$ of Dyck paths of length $2n$ with no peak at height 2 (mod 3) and no valley at height 1 (mod 3).

 Any non-empty peak-less Motzkin path with air pockets is either of the form (1) $H\alpha$  or (2) $U\alpha_1U\alpha_2\cdots U\alpha_k HD_k\beta$,  where $k\geq 1$ and $\alpha,\alpha_1, \ldots, \alpha_k,\beta$ are possibly empty peak-less MAP.  We refer to the left part  of  Figure \ref{fig2} for an illustration of this form.

Remember that $\mathcal{MP}$ denotes the set of  Motzkin paths with air pockets of the first kind.
\begin{defn} We  recursively define the map $\psi$ from  $\mathcal{MP}$ to $\cup_{n\geq 0}\mathcal{D}_n(2,1)$ as follows. For $\alpha\in\mathcal{MP}$, we set:
$$
\psi(P)=\left\{\begin{array}{llr}
\epsilon&\text{if }P=\epsilon&(i)\\
UD\psi(\alpha)&\text{if }P=H\alpha \text{ with } \alpha\in\mathcal{MP}&(ii)\\
U^3\psi(\alpha_1)DU\psi(\alpha_2)DU\ldots DU\psi(\alpha_k)D^3\psi(\beta)&\text{if }P=U\alpha_1U\alpha_2\ldots U\alpha_k HD_k\beta\text{ with}\\ & k\geq 1 \mbox{ and }\alpha_1,\ldots,\alpha_k,\beta \in\mathcal{MP}. &(iii)
\end{array}\right.
$$
\label{Def1}
\end{defn}
We refer to Figure~\ref{fig3} for an illustration of the third case of the definition of $\psi$.

\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=0.26]
\draw[very thick] (20,0) -- (21,1);
\draw[orange, very thick] (21,1) .. controls (22,4) and (24,4) .. (26,1) node[black,xshift=-20,yshift=6] {\large $\alpha_1$};
\draw[dashed] (21,1) -- (26,1);
\draw[dashed] (20,0) -- (47,0);
\draw[very thick] (26,1) -- (27,2);
\draw[orange, very thick] (27,2) .. controls (28,5) and (30,5) .. (32,2) node[black,xshift=-20,yshift=6] {\large $\alpha_2$};
\draw[dashed] (27,2) -- (32,2);
\draw[very thick]  (32,2) -- (33,3);
\draw[dashed] (33,3) -- (36,3);
\draw[very thick]  (36,3) -- (37,4);
\draw[dashed]  (37,4) -- (41,4);
\draw[orange, very thick] (37,4) .. controls (38,6) and (40,6) .. (41,4) node[black,xshift=-16,yshift=6] {\large $\alpha_k$};
\draw[very thick]  (41,4) -- (42,4)--(43,0);
\draw[orange, very thick] (43,0) .. controls (44,3) and (46,4) .. (47,0) node[black,xshift=-16,yshift=6] {\large $\beta$};
\end{tikzpicture}$\quad\Longrightarrow$
\begin{tikzpicture}[scale=0.26]
\draw[orange, very thick] (23,-7) .. controls (24,-4) and (25,-4) .. (26,-7) node[black,xshift=-14,yshift=6] {\smaller $\psi(\alpha_1)$};
\draw[very thick] (20,-10) -- (23,-7);
\draw[very thick] (26,-7) -- (27,-8)-- (28,-7);
 \draw[orange, very thick] (28,-7) .. controls (29,-4) and (30,-4) .. (31,-7) node[black,xshift=-14,yshift=6] {\smaller $\psi(\alpha_2)$};
 \draw[very thick] (31,-7) -- (32,-8)-- (33,-7);
 \draw[dashed] (33,-7) -- (36,-7);
\draw[very thick] (36,-7) -- (37,-8)-- (38,-7);
  \draw[orange, very thick] (38,-7) .. controls (39,-4) and (40,-4) .. (41,-7) node[black,xshift=-14,yshift=6] {\smaller $\psi(\alpha_k)$};
  \draw[very thick] (41,-7) -- (44,-10);
  \draw[orange, very thick] (44,-10) .. controls (45,-7) and (47,-7) .. (48,-10) node[black,xshift=-14,yshift=6] {\smaller$\psi(\beta)$};
 \draw[dashed] (23,-7) -- (26,-7); \draw[dashed] (28,-7) -- (31,-7);\draw[dashed] (38,-7) -- (41,-7);
\draw[dashed] (21,-9) -- (43,-9);\draw[dashed] (22,-8) -- (42,-8);
\draw[dashed] (20,-10) -- (48,-10);
\end{tikzpicture}
\caption{Illustration of the map $\psi$ for the more general case ($iii$) in Definition~\ref{Def1}.}
\label{fig2}
\end{figure}

Due to the recursive definition, the image  of peak-less MAP of length $n$ under $\psi$ is a Dyck path of length $2n$. Moreover it is clear that the obtained path is a Dyck path in $\cup_{n\geq 0}\mathcal{D}_n(2,1)$. For instance (see Figure \ref{fig3} for an illustration of this example).
\begin{align*}
\psi(UUHDHUHUHD_3HH)&=\psi(U\cdot \overbrace{UHDH}^{\alpha_1}\cdot U \cdot \overbrace{H}^{\alpha_2} \cdot U\cdot \overbrace{\epsilon}^{\alpha_3}\cdot H D_3 \cdot \overbrace{HH}^{\beta}\\
&=U^3\psi(UHDH)\cdot DU \cdot \psi(H) \cdot DU \cdot \psi(\epsilon)\cdot  D^3 \cdot \psi(HH)\\
&=U^3\cdot (U^3 D^3 \psi(H)) \cdot DU \cdot UD \cdot DU \cdot  D^3 \cdot UDUD\\
&=U^6D^3UD^2U^2D^2UD^2DUDUD.
\end{align*}

\begin{figure}[H]
\centering
\includegraphics[scale=0.75]{Ex2.eps}
\caption{$\psi(UUHDHUHUHD_3HH)=UUUUUUDDDUDDUUDDUDDDUDUD$.} \label{fig3}
\end{figure}

\begin{thm}
For all $n\geq 0$, the map $\psi$ induces a bijection between $\mathcal{MP}_n$ and $\mathcal{D}_{n}(2,1)$.
\label{bij}
\end{thm}
\noindent {\it Proof.}  Since $\mathcal{MP}_n$ and $\mathcal{D}_n(2,1)$ have the same cardinality (due to Corollary~2 and \oeis{A152171} in \cite{oeis}), it suffices to prove that for $P,Q\in \mathcal{MP}$, $P\neq Q$ implies $\psi(P)\neq \psi(Q)$. A simple induction on $n$ allows to obtain the result.
\hfill $\Box$

\section{Partial valley-less Motzkin paths with air pockets}

Using the same arguments we used for the system of the previous section, we study partial Motzkin paths with air pockets of the first kind avoiding occurrences of $D_iU$ for all $i\geq 1$.

\subsection{Enumerative results}

In the same way as we done in Section 2.1,  we have to solve the following system of equations:
 \begin{equation}\left\{\begin{array}{l}
f_0=1,\mbox{ and } f_k=zf_{k-1}+zh_{k-1}, \quad k\geq 1,\\
g_k=z\sum\limits_{\ell\geq k+1} f_\ell+z\sum\limits_{\ell\geq k+1} h_\ell,\quad k\geq 0,\\
h_k=zf_k+zg_k+zh_k,\quad k\geq 0.\\
\end{array}\right.\label{equ2}
\end{equation}
Multiplying by $u^k$ the recursions in (\ref{equ2}) and summing over $k$, we have:
\begin{align*}
F(u)&=1+zuF(u)+zuH(u),\\
G(u)&=\frac{z}{u-1}(F(u)-F(1)+H(u)-H(1)),\\
H(u)&=zF(u)+zG(u)+zH(u).
\end{align*}

Notice that we have $(1-z)F(1)=1+zH(1)$ by considering the first equation. Now, setting $f1:=F(1)$ and solving these functional equations, we obtain
$$F(u)={\frac {{\it f1}\,u{z}^{2}-u{z}^{2}+zu+{z}^{2}-u-z+1}{{u}^{2}z+{z}^{2}
-u-z+1}},$$
$$G(u)=-{\frac {{\it f1}\,uz+{\it f1}\,z-zu-{\it f1}+1}{{u}^{2}z+{z}^{2}-u-z+
1}},\quad
H(u)=-{\frac {z \left( {\it f1}\,uz-zu-{\it f1}+u+z \right) }{{u}^{2}z+{z}^
{2}-u-z+1}}.$$

In order to compute $f1$, we use the kernel method (see~\cite{ban, pro}) on $F(u)$. We can write the denominator (which is a polynomial in $u$ of degree 2), as $z(u-r)(u-s)$ with
$$r={\frac {1+\sqrt {-4\,{z}^{3}+4\,{z}^{2}-4\,z+1}}{2z}}\quad\mbox{ and } \quad
s={\frac {1-\sqrt {-4\,{z}^{3}+4\,{z}^{2}-4\,z+1}}{2z}}.$$

Plugging $u=s$ (which has a Taylor expansion at $z=0$) in $F(u)z(u-r)(u-s)$, we obtain the equation
${\it f1}\,s{z}^{2}-s{z}^{2}+zs+{z}^{2}-s-z+1 =0,$ which implies that $$f1=1+\frac{s-1}{z}.$$

Finally,  after simplifying by the factor $(u-s)$ in the numerators and denominators, we obtain
 $$F(u)=\frac{r}{r-u},\quad G(u)=\frac{s-1}{z(r-u)},\quad\mbox{ and } \quad H(u)=\frac{s}{r-u},$$
which implies that
$$f_k=[u^k] F(u)=\frac{1}{r^k}, \quad g_k=[u^k] G(u)=\frac{s-1}{zr^{k+1}},\quad \mbox{ and }\quad
h_k=[u^k] H(u)=\frac{s}{r^{k+1}}.
$$

\begin{thm} The bivariate generating function for the total number of partial valley-less MAP  with respect to the  length  and the height of the end-point is given by
$$\mathit{Total}(z,u)=\frac{s}{z(r-u)},$$ and we have
$$[u^k] \mathit{Total}(z,u)=\frac{s}{zr^{k+1}}.$$
Finally, setting $t(n,k)=[z^n][u^k]\mathit{Total}(z,u)$, we have for $n\geq 2$, $k\geq 1$,  $$t(n,k)=t(n,k-1)+t(n-1,k)-t(n-1,k-2)-t(n-2,k),
$$
and setting $t_n:=t(n,0)$ and $t_{-1}=1$, then we have for $n\geq 2$, $$t_{n-1}=t_{n-2}+\sum\limits_{k=0}^{n-3}t_{k-1}t_{n-k-4}+\sum\limits_{k=2}^{n-1}\left(t_{k-1}-t_{k-2}\right)t_{n-k-2}.$$
\end{thm}
\noindent {\it Proof.} The first three equalities are immediately deduced from the previous results. For the last equality, the term $t_n$ satisfies the same recurrence relation as in  Theorem 1 (modulo shift of $n$) since the two sequences are equal modulo a shift.
\hfill $\Box$

\begin{cor} The generating function that counts the partial valley-less MAP with respect to the length is given by
$$\mathit{Total}(z,1)=\frac{s}{z(r-1)}.$$
\end{cor}
The first few terms of the series expansion of $\mathit{Total}(z,1)$ are
$$1 + 2z + 5z^2 + 13z^3 + 34z^4 + 90z^5 + 242z^6 + 660z^7 + 1821z^8 + 5073z^9 + O(x^{10}),$$
which does not appear in \cite{oeis}.

\begin{cor} The generating function  that counts the partial valley-less MAP with respect to the length is given by
$$\mathit{Total}(z,0)=\frac{s}{zr}.$$
\label{cor5}
\end{cor}
The first few terms of the series expansion of $\mathit{Total}(z,0)$ are  $$1 + z + 2z^2 + 5z^3 + 12z^4 + 29z^5 + 73z^6 + 190z^7 + 505z^8 + 1363z^9+ O(z^{10}),$$ which corresponds to a shift of the sequence \oeis{A152171} in \cite{oeis} counting Dyck paths of length $2n$ with no peaks of height $2$ (mod 3) and no valleys of height 1 (mod 3) (see also Section 3.1).

Let $\mathcal{T}$ be the infinite matrix $\mathcal{T}:=[t(n,k)]_{n, k\geq 0}$. The first few rows of the matrix $\mathcal{T}$ are
\[\mathcal{T}=
\left(
\begin{array}{ccccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 2 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
 5 & 4 & 3 & 1 & 0 & 0 & 0 & 0 & 0 \\
 12 & 10 & 7 & 4 & 1 & 0 & 0 & 0 & 0 \\
 29 & 26 & 18 & 11 & 5 & 1 & 0 & 0 & 0 \\
 73 & 67 & 49 & 30 & 16 & 6 & 1 & 0 & 0 \\
 190 & 175 & 133 & 85 & 47 & 22 & 7 & 1 & 0 \\
 505 & 467 & 361 & 241 & 139 & 70 & 29 & 8 & 1 \\
\end{array}
\right).\]

In Figure \ref{fig3b} we show the valley-less PMAP counted by $t(4,0)=12$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.85]{Ex3.eps}
\caption{The 12 $UU$-less PMAP of length $4$ ending at height $0$.} \label{fig3b}
\end{figure}

\begin{cor}
The matrix $\mathcal{T}=[t(n,k)]_{n, k\geq 0}$ is a Riordan array defined by
$$\left((z^2-z+1)C\left(z(1-z+z^2)\right)^2,zC\left(z(1-z+z^2)\right)\right)$$
where $C(z)=\frac{1-\sqrt{1-4z}}{2z}$ is the generating function of the Catalan numbers
$c_n=\frac{1}{n+1}\binom{2n}{n}$.
\end{cor}
\noindent {\it Proof.}
 Indeed, we directly deduce the result from the following.
\begin{align*}
[u^k]\mathit{Total}(z,u)&=\frac{s}{zr^{k+1}}=\frac{s}{zr}\cdot \frac{1}{r^k}=(z^2-z+1)C\left(z(1-z+z^2)\right)^2\cdot\left( zC\left(z(1-z+z^2)\right)\right)^k .
\end{align*}
\hfill $\Box$

From a similar argument as in Corollary \ref{corformula} we obtain the following result.
\begin{cor}
We have
$$t(n,k)=\sum_{j=0}^{n-k}\frac{k+2}{2(n-j)-k+2}\binom{2(n-j)-k+2}{n-k-j}a(n-k-j+1,j),$$
where $a(n,k)=(-1)^k\sum_{i=0}^n\binom ni \binom{n - i}{k - 2 i}$.
\end{cor}

\subsection{A bijective approach}

Corollary~\ref{cor5} and Corollary~\ref{cor2} prove that the set of valley-less Motzkin paths with air pockets of length $n-1$ (ending on the $x$-axis) is equinumerous to the set of peak-less Motzkin paths with air pockets of length $n$, which is in one-to-one correspondence with the set $\mathcal{D}_n(2,1)$ of Dyck paths of length $2n$ with no peak at height 2 (mod 3) and no valley at height 1 (mod 3) (see a constructive bijection in Section~2.2). Below, we provide a bijection between valley-less MAP of length $n-1$ and peak-less MAP of length $n$.

 Any  valley-less Motzkin path with air pockets is either of the form ($i$) $\epsilon$, ($ii$) $\alpha H$,  ($iii$) $U\alpha D$, ($iv$) $\beta H U\alpha D$, ($v$) $U\gamma D_k$, or ($vi$) $\beta H U\gamma D_k$, where $\alpha,\beta$ are valley-less MAP (possibly empty), and   $\gamma D_{k-1}$ is a valley-less MAP.  According to all these cases, we define the map $\phi$.

 \begin{defn} We  recursively define the map $\phi$ from  valley-less MAP of length $n-1$ to peak-less MAP of length $n$. Let $P$ be a valley-less MAP, we set:
$$
\phi(P)=\left\{\begin{array}{llr}
H&\text{if }P=\epsilon,&(i)\\
\phi(\alpha)H&\text{if }P=\alpha H,&(ii)\\
U\phi(\alpha)D&\text{if }P=U\alpha D,&(iii)\\
\phi(\beta)U\phi(\alpha)D&\text{if }P=\beta HU\alpha D,&(iv)\\
\phi(\alpha D_{k-1})^\sharp&\text{if }P=U\alpha D_k,&(v)\\
\phi(\beta)\phi(\alpha D_{k-1})^\sharp&\text{if }P=\beta HU\alpha D_k,&(vi)\\
\end{array}\right.
$$
\label{Def1b}
where the $\sharp$-operator maps a peak-less MAP of the form $\alpha D_{k-1}$ into the peak-less MAP $(\alpha D_{k-1})^\sharp=U\alpha D_k$.
\end{defn}
\medskip

Due to the recursive definition, the image  of valley-less MAP of
length $n-1$ under $\phi$ is a peak-less MAP of length $n$. The recursive definition naturally induces that $\phi$ is a bijection. Using the bijection $\psi$ presented above, we can easily obtain a constructive bijection between valley-less MAP of length $n-1$ and Dyck paths of length $2n$ with no peak at height 2 (mod 3) and no valley at height 1 (mod 3).

\section{Partial $UU$-less Motzkin paths with air pockets}

In this section, we study partial Motzkin paths with air pockets of the first kind avoiding occurrences of $UU$.
\subsection{Enumerative results}
In the same way as we done in the previous section,  we have to solve the following system of equations:

\begin{equation}\left\{\begin{array}{l}
f_0=1, f_1=z+zg_0+zh_0, \mbox{ and }
f_k=zg_{k-1}+zh_{k-1}, \quad k\geq 2,\\
g_k=z\sum\limits_{\ell\geq k+1} f_\ell+z\sum\limits_{\ell\geq k+1} h_\ell,\quad k\geq 0,\\
h_k=zf_k+zg_k+zh_k,\quad k\geq 0.\\
\end{array}\right.\label{equ1}
\end{equation}

Multiplying by $u^k$ the recursions in (\ref{equ1}) and summing over $k$, we have:
\begin{align*}
F(u)&=1+zu+zuG(u)+zuH(u),\\
G(u)&=\frac{z}{u-1}(F(u)-F(1)+H(u)-H(1)),\\
H(u)&=zF(u)+zG(u)+zH(u).
\end{align*}

Notice that we have $H(1)=(1+z)(F(1)-1)$ by considering the first and the last equation. Now, setting $f1:=F(1)$ and solving these functional equations, we obtain
$$F(u)=\frac{\mathit{f1} u \,z^{3}+2 \mathit{f1} u \,z^{2}+u^{2} z^{2}-u^{2} z -2 u \,z^{2}+2 u z +z^{2}-u -z +1}{u^{2} z^{2}+u \,z^{3}+u z +z^{2}-u -z +1}
,$$
$$G(u)=-\frac{z \left(\mathit{f1} u \,z^{3}+2 \mathit{f1} u \,z^{2}-u \,z^{3}+\mathit{f1} \,z^{2}-u \,z^{2}+\mathit{f1} z +u z -z^{2}-2 \mathit{f1} +2\right)}{u^{2} z^{2}+u \,z^{3}+u z +z^{2}-u -z +1}
,$$
$$H(u)=\frac{z \left(\mathit{f1} u \,z^{3}+2 \mathit{f1} u \,z^{2}-u \,z^{3}+\mathit{f1} \,z^{2}-u^{2} z -2 u \,z^{2}+2 \mathit{f1} z +u z -z^{2}-u -2 z +1\right)}{u^{2} z^{2}+u \,z^{3}+u z +z^{2}-u -z +1}
.$$

In order to compute $f1$,  we use the kernel method (see~\cite{ban, pro}) on $F(u)$. We can write the denominator (which is a polynomial in $u$ of degree 2), as $z^2(u-r)(u-s)$, with
$$r={\frac {-{z}^{3}-z+1+\sqrt {{z}^{6}-2\,{z}^{4}+2\,{z}^{3}-3\,{z}^
{2}-2\,z+1}}{2{z}^{2}}} \
 \mbox{ and }$$
$$s={\frac {1-z-{z}^{3}-\sqrt {{z}^{6}-2\,{z}^{4}+2\,{z}^{3}-3\,{z}^{2}-
2\,z+1}}{2{z}^{2}}}
.$$

Plugging $u=s$ (which has a Taylor expansion at $z=0$) in $F(u)z^2(u-r)(u-s)$, we obtain the equation
$\mathit{f1} s \,z^{3}+2 \mathit{f1} s \,z^{2}+s^{2} z^{2}-s^{2} z -2 s \,z^{2}+2 s z +z^{2}-u -z +1=0,$ which implies that $$f1=1+\frac{s-1}{z(z+2)}.$$
Finally,  after simplifying by the factor $(u-s)$ in the numerators and denominators, we obtain
 $$F(u)=\frac{z-1}{z}+\frac{r}{z(r-u)},\quad G(u)=\frac{s+z}{r-u},\quad\mbox{ and } \quad H(u)=\frac{zr+1}{z(r-u)}-1,$$
which implies that $f_0=1$, $g_0=\frac{s+z}{r}$, $h_0=\frac{1}{zr}$ and
$$f_k=[u^k] F(u)=\frac{1}{zr^k}, \quad g_k=[u^k] G(u)=\frac{s+z}{r^{k+1}},\quad \mbox{ and }\quad
h_k=[u^k] H(u)=\frac{1+zr}{zr^{k+1}}.
$$

\begin{thm} The bivariate generating function for the total number of $UU$-less PMAP  with respect to the  length  and the height of the end-point is given by
$$\mathit{Total}(z,u)=\frac{1 + u z}{(r - u) z^2},$$ and we have $$[u^0] \mathit{Total}(z,u)=\frac{1}{z^2r},$$ and for $k\geq 1$
$$[u^k] \mathit{Total}(z,u)=\frac{rz+1}{z^2r^{k+1}}.$$
Finally, setting $t(n,k)=[z^n][u^k]\mathit{Total}(z,u)$, we have for $n\geq 2$, $k\geq 1$,

\begin{multline*}
    t(n,k)=t(n, k - 1) + t(n - 1, k) - t(n - 1, k - 1) - t(n - 2, k) \\- t(n - 2, k - 2) - t(n - 3, k - 1).
\end{multline*}
and setting $t_n:=t(n,0)$, then we have  $$t_n=t_{n-1}+t_{n-2}+t_{n-3}+\sum\limits_{k=2}^{n-2}t_{k-2}t_{n-k-2}+\sum\limits_{k=3}^{n-1}(t_{k-1}-t_{k-2})t_{n-k-1}.$$
\end{thm}
\noindent {\it Proof.} The first three equalities are immediately deduced from the previous results. Now, let us prove the last equality.
 Any non-empty length $n$ $UU$-less MAP is of the form ($i$) $HP$ where $P$ is a $UU$-less MAP of length $n-1$, or ($ii$) $UDQ$ where $Q$ is a MAP of length $n-2$ avoiding $UU$, or ($iii$) $U Q D R$, where $Q,R$ are some MAP avoiding $UU$ such that the length of $Q$ lies into $[1,n-2]$ and $Q$ starts and ends with $H$, or ($iv$) $P Q$,  where $P=UP'D_{i}$, $i\geq 2$, and $P'D_{i-1}$ is a MAP  of length lying into $[3,n-1]$ and starting with $H$. The number of $P'D_{i-1}$ of a given length $k$ is the total number of $UU$-less MAP of length $k-1$ minus the total number of $UU$-less MAP of length $k-2$. Taking into account all these cases, we obtain the result.

\hfill $\Box$

\begin{cor} The generating function that counts the partial $UU$-less MAP with respect to the length is given by
$$\mathit{Total}(z,1)=\frac{1+z}{(r-1)z^2}.$$
\end{cor}

The first few terms of the series expansion of $\mathit{Total}(z,1)$ are
$$1 + 2z + 4z^2 + 9z^3 + 21z^4 + 50z^5 + 122z^6 + 302z^7 + 759z^8 + 1928x^{9}+O(x^{10}),$$
which does not appear in \cite{oeis}.

\begin{cor} The generating function that counts $UU$-less MAP with respect to the length is given by
$$\mathit{Total}(z,0)=\frac{1}{z^2r}.$$
\label{cor11}
\end{cor}
The first few terms of the series expansion of $\mathit{Total}(z,0)$ are  $$1 + z + 2z^2 + 4z^3 + 9z^4 + 20z^5 + 47z^6 + 112z^7 + 274z^8 +  679x^9+O(z^{10}),$$ which corresponds to the sequence \oeis{A095980}  in \cite{oeis} counting Motzkin paths of length $n$ with no occurrences of $UHU$.

Let $\mathcal{T}$ be the infinite matrix $\mathcal{T}:=[t(n,k)]_{n, k\geq 0}$. The first few rows of the matrix $\mathcal{T}$ are
\[\mathcal{T}=
\left(
\begin{array}{ccccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 4 & 4 & 1 & 0& 0 & 0 & 0 & 0 & 0 \\
 9 & 9 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\
 20 & 21 & 8 & 1 & 0 & 0 & 0 & 0 & 0 \\
 47 &50 & 21 & 4& 0 & 0 & 0 & 0 & 0 \\
 112& 121 & 55 & 13 & 1 & 0 & 0 & 0 & 0 \\
 274 & 298 & 143 & 39 & 5 & 0 & 0& 0& 0\\
\end{array}
\right).\]
In Figure \ref{fig4} we show the $UU$-less PMAP counted by $t(4,0)=9$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.85]{Ex4.eps}
\caption{The 9 $UU$-less PMAP of length $4$ ending at height $0$.} \label{fig4}
\end{figure}

The matrix $\mathcal{G}$ is not a (proper) Riordan array. For this reason, we consider the matrix  $\mathcal{G}:=[g(n,k)]_{n\geq 0, k\geq 0}$, where
$$g(n,k)=\begin{cases}1, & \text{ if } n=k=0;\\
t(n+k-1,k), & \text{ if } n\geq 1.
\end{cases}$$

The first few rows of the matrix $\mathcal{G}$ are
\[\mathcal{G}=
\left(
\begin{array}{ccccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
 2 & 4 & 3 & 1 & 0 & 0 & 0 & 0 & 0 \\
 4 & 9 & 8 & 4 & 1 & 0 & 0 & 0 & 0 \\
 9 & 21 & 21 & 13 & 5 & 1 & 0 & 0 & 0 \\
 20 & 50 & 55 & 39 & 19 & 6 & 1 & 0 & 0 \\
 47 & 121 & 143 & 113 & 64 & 26 & 7 & 1 & 0 \\
 112 & 298 & 372 & 319 & 203 & 97 & 34 & 8 & 1 \\
\end{array}
\right).\]

\begin{cor}
The matrix $\mathcal{G}=[g(n,k)]_{n, k\geq 0}$ is the  Riordan array defined by $(t(z),t(z)-1)$, where
$$t(z)=\frac{1 + z(1-z)^2 -\sqrt{(1 - 3 z + z^3) (1 + z + z^3)}}{2 z(1 - z + z^2)}.$$
\end{cor}
\noindent {\it Proof.}
It follows from the relation $t(z)=\frac{1}{zr}+1$.
\hfill $\Box$
\medskip

It seems difficult to obtain a close form for the coefficient $g(n,k)$ using the Lagrange Inversion Formula. Indeed,  $t(z)$ can be expressed in terms of $C(u)$, where $u=\frac{1}{4}(-z^6+2z^4-2z^3+3z^2+2z)$ is a polynomial of degree 6, which complicates the calculations.

\subsection{A bijective approach}
Corollary~\ref{cor11} proves that the set of $UU$-less Motzkin paths with air pockets of length $n$ (ending on the $x$-axis) is equinumerous to the set of  Motzkin paths of length $n$ avoiding $UHU$. Below, we provide a bijection between these two sets.

 Any  $UU$-less Motzkin path with air pockets is either of the form ($i$) $\epsilon$, ($ii$) $H\alpha$,  ($iii$) $UD\alpha $, ($iv$) $UHD\alpha$, ($v$) $UH\alpha H D\beta$, or ($vi$) $UH^k\gamma D_i \beta$, where $\alpha,\beta$ are $UU$-less MAP (possibly empty), and   $\gamma D_{i-1}$ is a $UU$-less MAP and $k\geq 1$.  According to all these cases, we define the map $\chi$.

 \begin{defn} We  recursively define the map $\chi$ from  $UU$-less MAP of length $n$ to $UHU$-less Motzkin paths of length $n$. Let $P$ be a $UU$-less MAP, we set:
$$
\chi(P)=\left\{\begin{array}{llr}
\epsilon&\text{if }P=\epsilon,&(i)\\
H\chi(\alpha)&\text{if }P=H\alpha,&(ii)\\
UD\chi(\alpha)&\text{if }P=UD\alpha,&(iii)\\
UHD\chi(\alpha)&\text{if }P=UHD\alpha,&(iv)\\
UHH\chi(\alpha) D\chi(\beta)&\text{if }P=UH\alpha H D\beta,&(v)\\
U\chi(\gamma D_{i-1})H^{k-1} D \chi(\beta)&\text{if }P=UH^k\gamma D_i \beta,&(vi)\\
\end{array}\right.
$$
\label{Def3}
\end{defn}
\medskip

Due to the recursive definition, the image  of $UU$-less MAP of length $n$ by $\chi$ is a $UHU$-less Motzkin path  of length $n$. The recursive definition naturally induces that $\chi$ is a bijection. For instance, if  $P=UHUHD_2UDUHUHHUD_3H$ then we obtain $\chi(P)=\chi(UH^1UHD_2)\chi(UD)\chi(UHUHHUD_3)\chi(H)=UUHDDUDUUUDHDDH$ (see Figure \ref{fig5} for an illustration of this example).
\begin{figure}[H]
\centering
\includegraphics[scale=0.85]{Ex5.eps}
\caption{$\chi(P)=UUHDDUDUUUDHDDH$.} \label{fig5}
\end{figure}

redpajama_set_name: RedPajamaArXiv


text:
Loomis rankled Cleveland police's Black Shield union members on Trump, Browns, president says
Updated Jan 11, 2019; Posted Nov 21, 2017
Steve Loomis was defeated Tuesday in his re-election bid for president of the Cleveland Police Patrolmen's Association union. (Joshua Gunter, cleveland.com)
By Adam Ferrise, cleveland.com
CLEVELAND, Ohio -- Ousted Cleveland Police Patrolmen's Association President Steve Loomis angered members of the city's Black Shield Association police union over several issues related to President Donald Trump and the controversy surrounding Cleveland Browns players kneeling during the national anthem.
Cleveland's Black Shield Association president Lynn Hampton said some of his some 200 black officers represented by the organization expressed concerns over Loomis' decision to wade into those contentious debates.
"Their dismay may have come from some of the stuff that the CPPA weighed in on unnecessarily," Hampton said. "They felt we didn't need that kind of attention and that the negative press could have been avoided."
Hampton said Tuesday that he's not yet sure how many of his some 200 officers voted in the union president election that ended with Loomis losing to former police union president Jeff Follmer. The ballots are cast in secret over the phone.
Follmer beat Loomis by 38 votes out of 736 that were cast. There were 1,243 ballots sent out to union members.
Hampton said several officers expressed private reservations about Loomis leading the campaign to have the union endorse a presidential candidate in the 2016 election. The union had never before endorsed a president.
The turnout was low for the Trump endorsement vote with only 284 officers casting votes to endorse the president.
Hampton also said officers were upset after Loomis criticized the Browns for allowing players to kneel during the national anthem in protest of police brutality against mostly black citizens.
Hampton said he did not encourage voters one way or the other. He said both Loomis and Follmer reached out to the Black Shield to hear their concerns.
He said both promised more inclusion in the decision making of the CPPA, but that Follmer was "more specific."
Follmer, who at the time criticized the union holding the endorsement election, said in all future endorsement votes, he will allow officers to vote to endorse no one.
Follmer told Hampton that he learned a lot from some of his missteps he made during his previous two years as CPPA president, including when he was criticized for calling former Browns wide receiver Andrew Hawkins "pathetic" for wearing a "Justice for Tamir" shirt during pregame warm-ups before a 2014 game.
Loomis built a reputation as an outspoken pundit criticizing any criticism of police officers that followed many high-profile shootings of black men in communities throughout the country, including the November 2014 shooting of Tamir Rice by a Cleveland police officer.
He also endeared himself to the Trump campaign during the Republican National Convention and most recently traveled to Washington to attend the president's declaration of a national opioid crisis.
"We'll see how this goes moving forward," Hampton said. "It's an interesting time for our police officers and we need to be mindful that we need build trust with the community. We need to be aware we can do things to fan the flame."

redpajama_set_name: RedPajamaCommonCrawl


These reads–fresh picked for you from the Genuinely team–focus on connection. Whether it's finding your tribe, cultivating self-awareness and a positive attitude, using stories to bond, or practicing courageous vulnerability, these posts will provide additional insight on how to find and foster connection with your people.
Joel Gascoigne, Co-Founder of Buffer provides a transparent account of the current state of the company, the reason for their recent layoffs, and the changes they're making to recover. Having the courage to communicate and connect with their community on this level is the ultimate display of vulnerability and authenticity.
A long(er) read about how powerful stories can be for the leader who is looking to foster connection and build more fulfilling relationships with the people on his or her team.
You'll see it everywhere: employees want more than a paycheck. They want passion, connection, and a goal. Cultivating that work culture where employees feel comfortable enough to do that is an essential step, and Paul Alofs has eight easy tips to do it.
Just because your team is spread out across the country doesn't mean you can't cultivate your own workplace culture and forge essential working connections. Jeff Haden has four tips on how to do just that.


What makes you the same person as you were yesterday?
What makes you the same person as you were five years ago?
survive the total destruction of the body, including the brain.
In the Harry Potter books and films, memory charms are used, one of these being obliviate.
The charms are usually used to erase specific memories, for example, when a muggle (nonwizard) witnesses something that they shouldn't have.
photographs in the room. It is as if she never existed.
and identity. He ends up as a permanent resident of St. Mungo's hospital.
throws the stone when she has travelled back.
But what if things had changed?
Think about these two situations. Are the people the 'same' people afterwards?
rather it is consciousness, specifically memory.
the culprit. But he claimed that God would know the truth of our hearts.
person as the one who experienced that event?
really be a WWI soldier?
A 60-year-old man remembers fighting in the war when he was 25.
The 25-year-old man remembers going on holiday when he was 10.
But the 60-year-old man cannot remember going on holiday when he was 10.
60 can't be the same as the boy aged 10. Does this make sense?
memories, would you do so? Or do they make you who you are today?


Evinox has added to its CIBSE certified Technical and Design team, with Design Engineer Arturs Kokins becoming the latest to successfully complete the CIBSE Heat Networks Code of Practice course, passing his final exam with flying colours.
The CIBSE Heat Network qualification is recognised across the industry and covers a broad spectrum of considerations for communal and district heating systems, demonstrating the depth of knowledge Evinox can offer to their customers in the design of heat network systems.
Arturs originally joined Evinox in early 2017 as a Graduate Engineer, with an MEng (Hons) Mechanical Engineering degree, and has since developed his skills and gained valuable experience working on a number of key projects. These include managing bespoke testing with BSRIA (Building Services Research and Information Association) for a large development in London where Evinox is supplying heating and cooling units. He was also involved in Evinox successfully completing testing under the new BESA (British Engineering Services Association) UK Standard for Heat Interface Units, for which the Evinox ModuSat (XR) HIU achieved excellent results and is currently one of only five manufactures to have officially published its results on the BESA website, something the company is very proud of.
Evinox offers developers, consultants and specifiers a broad range of design support services for district and communal heating projects, including HIU sizing and selection; hydraulic system guidance and advice; and complete plant room system and distribution pipework design, where required.
Please contact Evinox on 01372 722277 or complete the contact form to find out more about our Design Services or BESA tested Heat Interface Unit range.


Efficient mailrooms are critical for many businesses, which is why IntelliTrack developed an innovative solution to help you manage your mailroom with ease. Our unique IntelliTrack software helps a wide range of organizations track their packages, including government agencies, universities, hospitals, and corporations manage large numbers of packages, multiple mailrooms, and loading docks.
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text:
Lacrosse tournament may cause traffic problems, Del Mar officials say
By Dave Rice, June 20, 2013
News Ticker alerts
The City of Del Mar is warning residents to prepare for traffic delays this weekend related to the LXMATS San Diego Lacrosse Tournament, which will take place at the San Diego Polo Fields off El Camino Real and is expected to draw up to 2,000 fans and participants.
"In this tournament elite level youth and high school teams will compete against each other and pro players will host clinics and demonstrations," says a release from the city.
The three-day tournament runs from 8 a.m. to 6 p.m. June 21-23.
Another event, the San Diego County Fair, is also occurring in the area over the same weekend. City officials say that the combination of the two may cause congestion on Interstate 5 in the Del Mar area, and are advising drivers to plan accordingly.


text:
Office of Worcester County District Attorney Joseph D. Early, Jr.
Meet Your DA
Drug Forfeiture Community Reinvestment
Community Events & Photos
Unresolved Case Squad
Litigation Integrity Division
Children's Advocacy Center
Nicholas Bisceglia - Fitchburg
Nicholas passed at the age of 26 after a 5 year battle with addiction, depression, and anxiety. He was a funny young man who could have done stand-up comedy. He had the ability to make those who knew him laugh with his infectious sense of humor. He was a talented barber with a large client base who would wait for him while he was away in treatment. Nick loved children, especially his nieces and nephew. He is sadly missed by his family and friends.
Share your thoughts and prayers:
» View Memorial Wall
» Submit a memorial of a loved one lost to the disease of addiction
info@worcesterda.org
@worcesterda
Worcester County District Attorney's Office
225 Main Street G-301 Worcester, MA 01608
Distracted Driving Resources
Opioid Task Force Community Prevention Programs
Treatment and Recovery Resources
Underage Drinking Prevention Resources
© 2020 The Office of the Worcester County District Attorney. All Rights Reserved.
The Office of the Worcester County District Attorney does not discriminate on the basis of race, color, national origin, religion, sex, disability, and age in the delivery of services or employment practices.


text:
Q: How to get rid of empty lines when using StreamWriter I'm trying to write a text file, from a datatable, text file which later on will be imported by some other application, therefor I need to have no empty lines in the exported file (at the end or between).
I used the following code:
DataTable table = FisierSoldata.dtSoldata;
 var result = new StringBuilder();
        foreach (DataRow row in table.Rows)
        {
            for (int i = 0; i < table.Columns.Count; i++)
            {
                result.Append(row[i].ToString());
                result.Append(i == table.Columns.Count - 1 ? "\n" : " ");
            }
            result.AppendLine();
        }

        StreamWriter objWriter = new StreamWriter(filePath, false);
        objWriter.WriteLine(result.ToString());
        objWriter.Close();

My datatable has 26 records and when the file is generated, it has one dataline followed by one empty line and another dataline and so on and at the end after the last dataline it has 3 empty lines.
How can I modify the code in order to have only datalines without the empty one between data and without the last 3 empty ones?
Regards,
LE:I've found a workaround, by reading some similar thread, but the solution was to use both streamwriter and streamreader in order to remove the unneeded data (empty lines) from the file which will be generated. But I wanted to know if there's a way of getting the file as needed from the start.

A: Why not using File.WriteAllLines, it's using a StreamWriter behind the scenes but allows simpler code:
var fileLines = table.AsEnumerable()
    .Select(row => String.Join(" ", row.ItemArray))
    .Where(line => !String.IsNullOrWhiteSpace(line));
File.WriteAllLines(filePath, fileLines);

But you really want to use a space as column delimiter? That's very error-prone. It will fail as soon as one field value contains a space.

A: I would try to avoid to add the empty lines to the Stringbuild. The Stringclass should give appropriate functions, maybe something like this:
            String stringToTestIsEmpty = row[i].ToString();
            //Check if it is not an empty line
            if( ! String.IsNullOrWhiteSpace(stringToTestIsEmpty ))
            {
                resutl.Append(stringToTestIsEmpty);
            }

redpajama_set_name: RedPajamaStackExchange


text:
Q: WARN (org.hibernate.jdbc.JDBCContext:333) - afterTransactionCompletion() was never called I am using hibernate as ORM when i checked my logs (catalina.out)
I am getting this warning so many times
WARN   (org.hibernate.jdbc.JDBCContext:333) - afterTransactionCompletion() was never called

Can any one please help me to solve this warning...?
what is the meaning of that warning and when it occurs ??

A: In the file JDBCContext.java source file is the following comment
/**
 * We cannot rely upon this method being called! It is only
 * called if we are using Hibernate Transaction API.
 */

This means, the method afterTransactionCompletion() normally is called by the hibernate transaction API, for example Session.flush() or Transaction.commit(). It seems somewhere in your program it can happen a transaction is not ended regularly. We can't tell you where, because we don't know your code.


text:
Q: I can't use tr() in Qt 5.5 outside the Qt classes I'm a little beginner, and I've a problem with implementation of the tr function in Qt. If I'm using tr("string") outside the Qt classes I'm getting errors.
I found information, that I should use QObject:: before the tr(), but If I try to do it with
temp += QObject::tr("Example"); // temp is std::string

I'm getting error
C2679: binary '+=' : no operator found which takes a right-hand operand of type 'QString' (or there is no acceptable conversion)

Another example:
    QString filename="example.txt";
    QFile log(QDir::currentPath() + "//" + filename);
    if ( log.open(QIODevice::ReadWrite | QIODevice::Append | QIODevice::Text) )
    {
    QTextStream plik( &log );
    plik.setCodec("UTF-8");
    (...)
    plik << QString::fromUtf8(QObject::tr("Example2")); // Error
    }

I'm getting error
C2665: 'QString::fromUtf8' : none of the 2 overloads could convert all the argument types

Could anyone help me with this issue?

A: Qt has so many accessors and QString::toStdString() as well.
temp += QObject::tr("Example").toStdString(); // temp is std::string

for the stream need to convert either to Utf8 byte array:
plik << QObject::tr("Example2").toUtf8(); // fixed

or even better it accepts QString as well.
plik << QObject::tr("Example2"); // will do


Browse featured blog posts, news stories, and updates below to learn more about the ins and outs of the Transcription Center and the volunpeer experience, the stories behind the digitized materials featured as projects, and the archival and museum staff that steward these Smithsonian collections.
Have a question about one of the posts on Marginalia or want to share your own volunpeer experience as a blog entry? Reach out anytime at transcribe@si.edu- we'd love to feature your #TCDiscovery!


text:
Jiří Kalous (* 6. Dezember 1964, in Chomutov) ist ein ehemaliger Eishockeyspieler und gegenwärtig Eishockeytrainer. Zuletzt war er bis 2022 Cheftrainer beim HC Kometa Brno.

Karriere
1970, begann er in seiner Heimatstadt Chomutov mit dem Eishockeyspielen. Zwischen 1970 und 1979 bereitete er sich erst in der Schülerliga, anschließend in der Juniorenliga auf den späteren Werdegang vor.  Während seiner Zeit auf dem Gymnasium in Chomutov spielte er als Stürmer für  VTŽ Chomutov in der 1. Liga der Tschechoslowakei. 1984 bis 1986 leistete er seinen Wehrdienst und spielte dabei für den Zweitligisten VTJ Sušice.

Hochschulstudium und Engagement in Österreich

Nach Beendigung seines Wehrdienstes begann er mit dem Studium der Trainingsmethodik mit der Spezialisierung auf Eishockey an der Fakultät für Sportpädagogik und Sportwissenschaft der Karls-Universität Prag. 1993 schloss er dieses erfolgreich mit dem Staatsexamen ab.

Während des Studiums spielte er drei Jahre lang (1986–1989) für den Eishockeyklub I. ČLTK Prag, bevor er in der Saison 1991/92 als Spieler nach Österreich zum Erstligisten EC Ehrwald wechselte. Eine Saison später wurde er Trainer des Clubs und führte ihn (1993) zum Titelgewinn in der Österreichischen Nationalliga. Zu diesem Zeitpunkt war er 26 Jahre alt.

Rückkehr nach Tschechien
1994 kehrte Kalous aus Österreich nach Tschechien zurück trainierte eine Saison den H+S Beroun HC. 1995 wechselte er zum HC Slavia Prag, wo er die folgenden 14 Jahre tätig war. Er begann als Trainer bei dem Junioren, mit denen er gleich im ersten Jahr (1995/996) den Titel in der Juniorenliga gewann. Er begleitete den HC Slavia Prag eine Saison als Co-Trainer, um ein Jahr später (1997/1998) die Junioren des HC Slavia Prag erneut zum Titel zu führen.
1998 bis 2003 war er Trainer der Tschechischen Nationalmannschaft der Junioren (Jahrgang 1983) und trainierte die U20-Auswahl des HC Slavia Prag. Mit der Nationalmannschaft der Junioren belegte er 2001 bei der U18-Weltmeisterschaft in Finnland den 4. Platz.

2002 war Kalous in der Extraligamannschaft von Slavia Prag an der Seite von Vladimír Růžička und Ondřej Weissmann vor allem für die Planung und Vorbereitung der Trainingsabläufe, die individuelle Vorbereitung der Spieler und das Konditionstraining verantwortlich. In der Saison 2002/03 feierten sie mit der Mannschaft den ersten Extraligatitel in der jüngeren Geschichte des Klubs. 2003/2004 und 2005/2006 führten sie das Team zum 2. Platz der Extraliga. Nach der Saison 2006/2007 verließ Ondřej Weissmann das Dreiergespann. Kalous und Růžička führten das Team in der folgenden Saison (2007/2008) erneut zum Titelgewinn in der Extraliga. Eine Saison später (2008/2009) belegte das Team dem zweiten Tabellenplatz.

Nach seinem Engagement beim HC Slavia Prag wechselte Kalous zur Saison 2009/10 als Cheftrainer zum Extraligaverein Bílí Tygři Liberec. Er betreute die Mannschaft drei Jahre bis zum Ende der Saison 2011/12 und erreichte mit ihr jeweils die Play-offs, zweimal davon das Halbfinale.

Zur Saison 2012/13 wechselte er zum neu gegründeten Prager KHL-Team HC Lev Prag und wurde Co-Trainer von Josef Jandač.
Nach einem erfolgreichen Saisonstart kam es zu einer Serie von Misserfolgen, die zur Abberufung von Cheftrainer Josef Jandač führte. Kalous blieb beim Team und wurde zwischenzeitlich Cheftrainer des Teams, bevor Václav Sýkora das Team übernahm. Anfang 2013 beendete Kalous sein Engagement beim HC Lev Prag.

Ab der Saison 2013/14 war er Cheftrainer von HC Oceláři Třinec, ehe er im November 2015 abberufen wurde. Zur Saison 2016/17 wurde er Cheftrainer beim HC Sparta Prag. Parallel dazu war er ab 2015 Assistenztrainer bei der tschechischen Nationalmannschaft. Im Oktober 2017 wurde er vom HC Sparta Prag entlassen, betreute aber bis 2018 weiterhin das Nationalteam.

Zwischen 2018 und 2020 arbeitete Kalous als Co_Trainer beim HK Metallurg Magnitogorsk aus der KHL, anschließend war er ohne Anstellung. In der Saison 2021/22 war er Cheftrainer des HC Kometa Brno und Assistenztrainer beim tschechischen U20-Nationalteam.

Erfolge

 1993 Meister der  Österreichischen Nationalliga mit dem EC Ehrwald (als Trainer)
 1996 Meister der Junioren-Extraliga mit dem HC Slavia Prag (als Trainer)
 1998 Meister der Junioren-Extraliga mit dem HC Slavia Prag  (als Trainer)
 2003 Tschechischer Meister mit dem HC Slavia Prag (als Assistenztrainer)
 2004 Tschechischer Vizemeister mit dem HC Slavia Prag (als Assistenztrainer)
 2006 Tschechischer Vizemeister mit dem HC Slavia Prag (als Assistenztrainer)
 2008 Tschechischer Meister mit dem HC Slavia Prag (als Assistenztrainer)
 2009 Tschechischer Vizemeister mit dem HC Slavia Prag (als Assistenztrainer)
 2015 Tschechischer Vizemeister mit dem HC Oceláři Třinec (als Cheftrainer)

Weblinks

Eishockeyspieler (Tschechoslowakei)
Eishockeyspieler (Piráti Chomutov)
Eishockeytrainer (HC Slavia Prag)
Eishockeytrainer (Bílí Tygři Liberec)
Eishockeytrainer (HC Lev Prag)
Eishockeytrainer (HC Sparta Prag)
Eishockeytrainer (HC Oceláři Třinec)
Person (HK Metallurg Magnitogorsk)
Eishockeytrainer (HC Kometa Brno)
Tschechischer Meister (Eishockey)
Tscheche
Tschechoslowake
Geboren 1964
Mann

redpajama_set_name: RedPajamaWikipedia


text:
Q: Navigation View button on all screens and refresh home I'm creating an application to which I've added a Botton Navigation View but I don't know if there is any way that it appears in all the Activity or if on the contrary I would have to specify it in each of them to appear, as well as adding all its functionality on each screen, since this last option I see a little cumbersome.
The other question is for the button Home of this bar, I have added a button that is home to return to the main Activity each time I pulse, in the way that I have done has been with an Intent that goes to that screen itself, but I do not know if there is another method:
bottomNavigation.setOnNavigationItemSelectedListener(new BottomNavigationView.OnNavigationItemSelectedListener() {
        @Override
        public boolean onNavigationItemSelected(@NonNull MenuItem menuItem) {

            switch (menuItem.getItemId()) {

                case R.id.btnHome:

                    Intent intent = new Intent(MainActivity.this, MainActivity.class);
                    startActivity(intent);

                    break;

A: Check out how to use Fragment in Android. You don't have to open a new activity each time the button in your BottomNavigationView is clicked, instead, you use fragments, so only the selected part in activity is changed. Check out the following tutorial to learn more about Fragments. See the "dynamic creation" of fragments.
How to use Fragments in Android.
In your case, there'll be code of FragmentTransaction where you're launching a new Activity using Intent. You'll be replacing only a certain portion of the screen(between your Toolbar and `BottomNavigationView).


LOCATION! LOCATION! LOCATION! What an incredible villa in a beautiful setting with lots of trees and green space!! This is such a great location since it is just so close to everything including within walking distance to all of the great shows at Ruth Eckerd Hall, and in very close proximity to 'quaint downtown Safety Harbor, the airport, restaurants and lots of shopping! This one story unit boasts two very spacious (split) bedrooms with lots and lots of closet space, two updated bathrooms , volume ceilings, a remodeled enclosed Florida room, and a large eat-in kitchen with breakfast bar. The enclosed Florida room offers an additional 200 sq ft of living space w/ tile flooring, A/C and French doors which could be used as an office, den or bonus room, so this unit appears much larger than the tax records indicate. Other features include a utility closet for extra storage, plus a covered carport. The community provides some additional guest parking, community pool, a recreation building , fitness center and tennis courts!! Available for a fast closing, so don't miss out on a great opportunity. Make an appointment today!


text:
Sony: Not enough demand for UMD Passport service in U.S.
Sony has explained that the UMD Passport scheme for PS Vita is not coming to the U.S. or Europe, as there is simply not enough demand for the service, and PSP games are available from PSN for "reasonable" prices.
Sony has explained that the UMD Passport scheme for PS Vita is not coming to the U.S. or Europe, as there is simply not enough demand for the service. The UMD Passport program was announced last year as a method for PSP owners to bring their previously-purchased UMD games to the PS Vita handheld. However, while the service launched in Japan at Vita launch, it was revealed that it would not be available to either U.S. or European Vita users. SCE's head of worldwide studios Shuhei Yoshida has now told Wired that "there is a much larger demand for PSP games" in Japan, and other regions will not be receiving the service. "When you look at the release schedule of new titles there are still lots of PSP games being released in Japan and being announced for release," he explained. "Lots of people who are interested in trying Vita are also interested in playing PSP games that they might purchase before Vita comes out, and will not necessarily choose the digital version. So there is a lot more demand... to introduce a program like that." He also noted that PSP games are sold via the PSN store for U.S. and European users at "really reasonable" prices, meaning there is no need for the UMD Passport service to be brought across. "There are many, many games that are sold at an affordable price. Because people in Japan are not getting the digital copy for free, because it costs us money to develop and maintain the system so we are asking people to pay somewhere between $5 and $10 to receive the digital copy in addition to what they have on the UMD," he said. He finished, "When you compare that to the price of games here, PSP games in Japan are sold at a much higher price, so people see the value in spending the $5 to $10 to get the digital copy. But when the games are already sold at a lower price in the U.S. we see less value in introducing that kind of system."


Taking a vacation may seem like the most unproductive thing in the world, but in reality it can be one of the best things you ever did for yourself as well as your business. It is no secret that taking time to unplug and recharge can improve your physical, mental, and emotional health. Many do not realize however, that there can be excellent benefits to the business when you simply walk away for a short time.
A successful business should be able to run without you there. You may think that it can, but the only way to find out if it will is to truly walk away and be as unavailable as possible. When you do this, any glitches in the system, whether human or otherwise, will quickly come to light. While there may be some temporary chaos, this is actually a good thing. There is no way to repair a problem when you have no clue where it is.
The cream is the best part, and it always rises to the top. Not only will stepping away allow flaws to be found, but it will get you out of the way so that the cream of the crop can make itself known. You will be able to see who can and will step up to the plate. Some will continue to work as always, and if they always work well it will be good to know that they will do their best whether you are physically present or not. Others, however, will go above and beyond to ensure things run smoothly while you are away, and who it is may surprise you. You could discover an asset your never knew you had, and you never would have known had you not stepped away.
Sometimes it is hard to see changes that need to be made because you are too close to things or too busy to think clearly. If you do not ever slow down enough to process things, you could miss a lot as far as what is working and what isn't. Though you do not want to spend too much vacation time thinking about business, you definitely should allow yourself a little time to process and review how things are going and consider what may or may not be working.
All of that recharging, renewing, and relaxing you should be doing really can improve your ability to manage. It will increase your ability to put things in perspective and help you to better see things from others' point of view.
It may seem that the worst thing you could do for your business is walk away. No one is asking you to disconnect completely or to take an extended vacation leaving your baby to the wolves. There are definitely benefits however to leaving for a little while, and a little R&R is only one of them.


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Chris & Nick Spicer '93 & '96: Creativity + Business
According to the Spicer brothers, the teachers at Villa "challenged us to take risks and to think independently. It wasn't just about learning a curriculum, but about nurturing curiosity and fostering a love of learning."
Chris '93 (left), received his B.B.A. and B.A. from the UW, and then his J.D. from Georgetown University Law Center. Nick '96, majored in History and French at Rice University as a National Merit Scholar and later earned an MFA from the UCLA Producers Program in 2006.
Today, both brothers work in the film industry, at the intersection of art and business. Chris is a film-finance lawyer in LA at the Akin Gump firm and has helped provide financing for shows like House of Cards, and Orange Is The New Black, as well as for movies like The Hunger Games and Ender's Game. Nick is a founding partner at XYZ Films in LA and has produced films like The Raid I & II, Tusk, The Invitation, and The Oath.
They say, "We're fortunate in that the success of our projects, clients and deals are not measured solely in financial terms, but in how effectively a movie transports the audience emotionally."


The gift of holiday cheer… one squeaky toy at a time!
Learn how to support the Mercer Clinic Holiday Pet Basket and Winter Coat program here!
A group of volunteers at the UCD Veterinary Medical Teaching Hospital (VMTH) has carried on a tradition of making holiday pet "baskets" for the Mercer Veterinary Clinic for the Pets of the Homeless since 1995. The pet basket tradition stemmed from a VMTH staff newsletter meeting (The VOICE) where the group discussed how they could help with a holiday charity project. Marcia Nelson and Becky Griffey came up with the great idea of making holiday pet baskets for the pets of the homeless through the Mercer Veterinary Clinic associated with the UC Davis Veterinary Medical Teaching Hospital. This year we are celebrating our 28th year doing the Holiday Pet Basket event.
The Mercer Veterinary Clinic was started by vet students Randy Singer and Dori Borjesson in1992. It was modeled on the Doney free pet clinic in Seattle that they had heard about at the time. The clinic was named after a fictional character with an ability to share an empathetic response. The Mercer Pet Clinic is a non-profit organization which offers free veterinary care monthly, for the pets of the homeless and is operated by volunteer veterinary faculty, residents, and students from the UC Davis Veterinary School and relies entirely on donations. To this day, the Clinic is still held every month at the Loaves and Fishes complex at 1321 West "C" St. in Sacramento.
The need for the holiday pet basket program was evident to all of us who saw the photos of the first event, where the delighted pet owners and their pets seemed to be having a great time. It was clear from the feedback and the photos that these very special pets were more like family, than a pet, to their loving owners. Our group was moved and motivated to make this an annual event. Although these less fortunate folks did not have the means to give gifts to their beloved pets, we could help by continuing to organize this event annually, and making the holidays a little brighter for them.
Traditionally, the VMTH newsletter staff wrap the pet "baskets" (i.e. open empty boxes) with holiday wrapping before a noontime "Holiday Pet Basket Stuffing" gathering. Original VMTH newsletter members who still participate annually include Becky Griffey, and Eileen Samitz. Becky Griffey was the coordinator between 1995 until 1998 and Eileen Samitz has been the coordinator since 1998. These pet basket "elves" will be joined by other helpers, including UCD current or former staff members Becky Griffey, Judy Wall, Marion Derby, Monica Kratochvil, Diane Nayden, Jeanne Brown, Barbara Fowler, Felicia DonRussello, and Matt Nichols. We are grateful Pat Bailey and Trina Wood for helping us coordinate with the media to share this event with the UCD staff and the public. We are also ever so grateful to Celeste Borelli and Beatrice Le for their kind help in processing the donations.
Veterinary student "elves," Mercer Clinic officers, also help us with assembling, transporting and distributing the holiday pet baskets. These Mercer Officers themselves are regular volunteers at the Mercer Clinic during the school year. We appreciate the extra time that they invest in the pet basket program each year; the event has been an inspiration and a joy to all involved.
In the first year, the holiday pet basket event produced 25 baskets. In subsequent years, the number increased to 40, then 75, 90, 100, 110, 120 and eventually 130 last year. This year our goal is to make 130 baskets again. The pet "baskets" are sorted by being intended for a dog or cat and labeled accordingly, then filled with a variety of gifts such as food, treats, and of course, toys for the pets. In 2012 we voiced the need for sweaters and coats for the dogs to have warmer winter and we were fortunate enough to raise the money to provide sweaters and/or coats for the dogs who came to the Mercer Clinic in the winter months. We provided sweaters to a few cats of Mercer clients who would tolerate wearing them! Since we now yearly use up our entire supply of coats and sweaters, we are in great need of generous donations to continue this coat and sweater program for these special pets to help them weather the cold and wet elements that they face in our region.
The ability to provide the Mercer Clinic holiday pet baskets relies entirely on the generosity of donations from animal lovers who wish to help these needy pets, and pet supply companies who are kind enough to donate as well. Last year we were inspired by the support we received as we went about the difficult task of raising enough donations to fully fund and supply the program. However, we were fortunate enough to see generous contributions from several companies, as well as donations from VMTH staff due to word traveling throughout the VMTH. Hearing about our need, our terrific VMTH staff, plus other pet lovers outside of the VMTH, responded immediately with their kind supply of money and pet items. The staff of VMTH is known for their love of animals, reflected on a daily basis by their dedication to help and heal the animals that come to the VMTH. Once again, they are demonstrating their commitment to caring for animals, by helping the pets of the homeless this year.
Although the baskets are distributed at Christmas time, Mercer Clinic welcomes donations all year round, since it relies primarily on donations to operate. We have been fortunate enough to have been given pledges of donations or discounts in the past from companies such as Purina-Nestle, Hills/ Science Diet, Virbac., Butler - Schein Animal Health Supply, Pet Edge, UPCO, Dechra Veterinary Products, Merial, Merck, M.W.I, and Elanco, but the generous support of individual contributors provides the lion's share of the program's very lean budget.
Several years ago we learned that, unfortunately, the pets are the homeless are generally not allowed into the homeless shelters with their owners. So, we added the Mercer Winter Pet Coat and Sweater Program to help these special pets get through the cold and wet winters. This important program has helped many of pets of the homeless, particularly older or more frail pets, survive the winters. Last year we issued over 250 pets coats and 250 sweaters; however, we need even more generous contributions to continue this program, since it has become vital to the survival of these special pets.
As one might imagine, in order to produce 130 baskets filled with gifts, and also provide our winter coat distribution program, it takes many contributors to make it a success. Thanks to all the above participants we had the ingredients to produce the biggest and best holiday pet basket event ever, last year. However, we wish to remind everyone that every year we rely on the kindness of the public to help these very special pets who are more than a pet. Instead, they are the constant companions and family members to the less fortunate.
To make a financial donation (which is tax deductible) the Mercer Clinic Holiday Pet Baskets and Coats Program, please make your check to: UC Regents - Mercer Clinic Holiday Pet Baskets and send it to UC Davis School of Veterinary Medicine, Office of the Dean, P.O. Box 1167, Davis CA 95617-1167, Attn. Mercer Holiday Pet Baskets. Please include a donation form printout, which can be downloaded here.
Alternatively, you can make a donation for the Holiday Pet Baskets on the web at the UC Davis Give programs link: https://give.ucdavis.edu/Go/MercerVet.
All donations are tax-deductible and all donors receive a confirmation receipt for any monetary donations.
Mercer Veterinary Clinic and the crew of the Holiday Pet Baskets and Winter Pet Coat and Sweater program wish to thank all of our financial contributors for your generosity. Thanks also to all the volunteer participants that help us to continue these two programs. We also wish everyone a wonderful and happy holiday season.
Eileen M. Samitz, MT, (ASCP), M.A.
Mercer Clinic Holiday Pet Baskets and Winter Coat Program Donations Page.
Downloadable form for donations by mail (click here).
Mercer Veterinary Clinic for Pets of the Homeless has been proud to receive a number of awards including: the Biberstein Social Action Award in 2009 and 2015, the Outstanding Community Service Award in 2013 and 2015, the UC Davis Civic Engagement Award in 2015, and the Presidential Award for Community Service in 2015 from President Obama.
You will find more information about the year round services that are provided by the Mercer Veterinary Clinic for the Pets of the Homeless through our website here and our facebook page.
The year-round operations of the clinic are separately funded from the Holiday Basket program. We highly encourage you to be a part of our pet basket funding for this year, but a donation to the clinic itself would not go amiss as well! For anyone who wishes to make a financial donation to the Mercer Vet Clinic for the Homeless please have a look at our direct donation page here. Either way (or if you choose to give to both!) we appreciate you sharing a part in our mission to deliver health and wellness to the pets and people of our hometown.
Images on mercerclinic.com all come from actual clinic events. Images courtesy of the Mercer Clinic student historians, used with permission.


ORLANDO, Fla., Feb. 28, 2019 - On May 18th and 19th, Mouseketeers will reunite for a once-in-a-lifetime weekend with their fans at Walt Disney World® Resort and MEGACON™ Orlando to celebrate their shared experience with one of Disney's most iconic brands, and to give back to where it all began. Hosted by Joey Fatone, who grew up with the Mouseketeers in Orlando, proceeds from #MMC30 will benefit Central Florida's onePULSE Foundation and their efforts to promote diversity and inclusion, and Give Kids The World Village, a nonprofit resort for critically ill children that Walt Disney World® has supported for more than 30 years.
August 24, 2017 – Dr. Phillips Center for the Performing Arts announces the four local area elementary schools that have been selected to participate in the Disney Musicals in Schools program, now in its second year locally. New participants for the 2017 – 2018 school year are Kaley Lake Como Elementary (Orange County), OCPS Academic Center for Excellence (Orange County), Pleasant Hill Elementary (Osceola County) and Wicklow Elementary (Seminole County). Returning from last academic year for their second year in the three-year program are Bonneville Elementary (Orange County), Winegard Elementary (Orange County), Lake Orienta Elementary (Seminole County) and Kissimmee Elementary (Osceola County). At an orientation event in December 2017, these schools will select a Disney musical of their choice from the "KIDS" licensing collection. The rehearsals will officially start in January 2018, culminating with performances the following May.
(Friday, May 5, 2017 -- Orlando, FL) – The Orlando Philharmonic Orchestra is hosting a fundraising event presented by the Walt Disney World Resort on Tuesday, May 23 at DoveCote Brasserie at 6 p.m. The goal is to raise $100,000 to support the orchestra's season of concerts and education programs.


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As Coisas do Campo (Res Rusticae) é uma obra didática escrita pelo erudito romano Marco Terêncio Varrão (Marcus Terentius Varro), contemporâneo de Júlio César (portanto, do período final da República Romana). É a única obra deste prolífico polímata que chegou completa até nós.

Traduções e estudos em língua portuguesa
Em 2012, foi publicada a primeira tradução brasileira da obra, feita por Matheus Trevizan

Ligações externas
 O texto original em latim na Latin Library

Livros da Roma Antiga
Literatura latina


We are proud to be working with NOW Live Events at Deptford Lounge, London as part of the London-wide 'Anxiety Arts Festival' over June 2014. Our workshop is 4pm-4:45pm on Tuesday 24th June. Sarah is also making soft sculture bunting to decorate the venue with words on wellbeing. They will be up to see for the entire festival.
We will be part of this jam-packed programme championing an innovative blend of arts and wellbeing activities for all ages to stimulate the senses, lighten the mind and savour the moment with some of UK's most inspiring artists and thinkers, locally and beyond.
THere will be literature, performance, craft, poetry, visual arts, spoken word, music, mindfulness meditation, workshops for children and more. The programme includes Britain's leading lifestyle philosopher Roman Krznaric, spoken word artist Simon Mole and Chill Pill Collective, mindfulness expert Tessa Watt and us, Craftivist Collective.
All events will be FREE, apart from ticketed evening events on Monday, Thursday and Friday.
We often reflect on the time to come or time gone by without really enjoying where we are now. We are connected to others and we have an impact as a local and global citizen (for most of us quite a daunting thought). It is easy to let life pass us by so why not craft yourself a physical footprint to help you focus yourself on the NOW and remember how to be your best self in your local community using your unique passions, talents & actions?
In this workshop led by our Founder of Craftivist Collective Sarah Corbett, you will learn the classic skill of embroidery whilst also learning how to be a change-maker to make the world a better place for people and planet. You will create a fabric footprint for you to keep and you will receive support on what message to stitch on your footprint and suggestions of where to put it.
I'm looking forward to creating a space and environment for participants to feel safe and open to how my approach to craftivism can benefit them personally, collectively and politically.
Politics is personal and so is the solution. I hope people leave feeling they know more about the different benefits Craftivism can offer and how they can use it in their life and work & become craftivists!


A 53 year old Auxvasse man is in custody after deputies conducted a wanted person investigation in the 100 block of First Street in Auxvasse.
Deputies located Dewayne Clayton. He was wanted out of Boone County for probation violation. He was also allegedly found to in possession of methamphetamine.
He is being held at the Callaway County Jail.


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the third terminator Jan. 20, 2012
NYC Wants Single Adults Out of Homeless Shelters
By Joe Coscarelli
Photo: Spencer Platt/Getty Images
A proposal by New York City's Department of Homeless Services aims to free up space in local homeless shelters by sending single adults back to the last place they lived, lest they risk being forcibly removed from shelters. If there's a problem with space at the person's previous location, the DHS will provide furniture or even help rearrange the space to create more room. "There are modest changes that can be made to the layout of an apartment to accommodate another single," said DHS Commissioner Seth Diamond. "We think that that's appropriate." The City Council and Legal Aid Society of New York are suing the city over what they call a "dangerous" and "misguided" plan.
The Daily News reports:
Under the policy, tenants evicted by their landlords would not be eligible for shelters until a sheriff or marshal booted them.
"If the housing is still available, the applicant will be expected to retrieve their previously submitted keys from the landlord," the policy states.
The city would also deny shelter even when a friend or family member decides the person can't live with them anymore. Shelter applicants who can't provide documents proving they are homeless would also be ineligible.
City Council speaker Christine Quinn said, "I'm confident that the courts will rule in our favor and that the administration ultimately will be forced to abandon this irresponsible policy." A representative from the Coalition for the Homeless added, "It basically creates a mechanism for the city to deny shelter to people even when they have no place to go."
But Diamond insisted, "The objective is to reserve shelter space for people who have no other options."
HuffPo NYDN
the third terminator
the homeless
e jean carroll
E. Jean Carroll 'Exploring' Lawsuit Against Trump After Rape Allegation
Carroll said if she hires "a really smart attorney, we might be able to get around" the five-year statue of limitations for rape cases prior to 2006.


Interior first floor plan home building furniture house, interior design floor plan sketches datenlaborinfo. 27 floor plans for living room arranging furniture, kolea. Beautiful interior design and floor plan drawing home.
Set top view for interior icon design floor plan stock, furniture placement archives stellar interior design. Cafe and restaurant floor plan solution conceptdrawcom. House floor plan symbols escortsea.


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Kubernetes Dashboard
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redpajama_set_name: RedPajamaGithub


Mother.Domains was born in 2014 after inspired world travels and 20 years experience online.
Upon the Internet's advent of hundreds of new domain name extensions, there were infinite possibilities for new ideas, ventures and projects.
Mother.Domains seeks to help brainstorm, build and broadcast domain name-related ideas, providing comprehensive resources for the DIY types, and offering high end products & services for those who want a helping hand.
Based in Melbourne, Australia, with a growing worldwide network, every decision we make has an embedded social and environmental consciousness, as we see a pathway for domains, ideas and entrepreneurship making a lasting, positive impact on our world.
For more information about Mother.Domains, check out our website at https://mother.domains or contact us directly.


"Four snakes" is a project about nature and human settling in it. This action have to be light, quiet and sober. Nature life must continue as if there isn't any intrusion. To achieve this goal, the project develops as four essential building along the river, each of them dedicated to a specific function. People are allowed to walk over and below blocks , which are conceived as spaces of nature, not closed areas. The main objective is to overstep classic division between built up and external world, merging this two components in a unique well liveable conception.


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Q: NPC Thieves in Skyrim I have a chest in Riverwood where I'll keep everything I can't carry I.e. Dragon bones, books etc. And lately I've noticed an abundance of theives running away from riverwood whenever I fast travel there. I manage to run them down but this never happens in any other town.
Question:
Should I be worried that they are stealing from me?

A: No, NPC thieves won't break into your chests, but yes, items you leave unattended in chests may expire. The only safe chests are ones in a house you own. You can buy a house in Whiterun for 5,000 gold.

A: Any container that doesn't belong to the player, IE is not in a player-owned house can be considered unsafe for storing gear. If you leave stuff in a container you don't own, chances are it'll vanish after a few game-days. No-one is stealing it, just the contents of most containers are refreshed every so often. The new contents are selected by the game from a predefined list. Anything not on that list wont appear in the container when the contents are refreshed. So say you stored a unique sword in a random barrel somewhere. The sword will most likely disappear after either 24 hours or 3 days (the standard pre-set refresh intervals). Next time you visit the barrel, there might only be apples in it. Sometimes your item might survive a refresh and still be there when you look back... but don't count on it.
Player-owned containers do not have their contents refreshed, hence the stuff you leave in it will stay there until you remove it yourself. If things are missing from a player-owned container, it's either because you forgot that you moved them somewhere else or due to a bug in one or more of your saved games.
As for NPC theives; Some people believe that NPCs exchange objects with containers as a matter of course. They don't. An NPC can only exchange items in & out of a container if they are specifically scripted to do so. 99% of NPCs are not scripted to do this. The remaining 1% constitutes quest-specific NPCs (who only exchange pre-defined items from pre-defined containers) and traders. Traders do not normally exchange items with containers that are visible in the game world. Most have their own special containers which are positioned outside of the bitmap where they can't be seen. Sometimes it is possible for the player character to enter "Blue Hell" (fall outside the bitmap), where if you search carefully, you might find one of these special trader chests. Take care what you steal from these chests if you ever find one. Anything you take will never appear in that traders barter menu again. This includes gold. If you steal all the gold from a traders secret chest he'll never have any to trade with, unless you put gold back in the chest again.
Regards to NPCs found in the players home; no, they are not stealing anything. If an NPC is speaking to you as you enter a door, they will often spawn into the new cell with you to finish speaking their line. Sometimes the game will have difficulty knowing where in the cell to spawn them, in which case you may find them... in your bedroom, say.
Most times they will leave straight away. Some NPCs do wander in & out of buildings at random. Although player homes aren't strictly part of the AI wander package (the thing that controls where an NPC can & can't go), the thing with random generators is that they're random. An NPC with a random wander package can effectively go anywhere within the parameters defined by that package. For example, an NPC with a wander package specifically for "Whiterun, Dragonsreach" may enter unlocked cells within Dragonsreach only. An NPC with a wander package for "Whiterun" may enter any unlocked cells within Whiterun, including dragonsreach (and the players home). Wander packages do not negate locked doors, meaning that NPCs can't wander in & out of locked rooms or buildings unless or until the door is unlocked by the player or by a script.
When things in your house get strewn about the place... this isn't because an NPC wandered in and threw all your cabbages on the floor! It's just a glitch in the game. If things are missing they probably fell through the bitmap or were respawned outside of it thanks to a bug in your save game. When items fall through the bitmap they "C.O.W"... go to the Centre Of World. When this happens it's tough luck I'm afraid. You've lost that thing. Either learn the objectID and give it back to yourself using the command console or reload a saved game & hope the bug that caused the problem wasn't present when you saved it.
NPC theives? Yes, there are thief characters but no, they don't actually steal anything. Why they're appearing for you in Riverwood is anyone's guess. Random spawners, saved game bugs, who knows. These things happen!
Most recently, I entered my house in Whiterun to find Lydia and Uthgerd the barbarian stood either side of a seated Faendal. All three butt-ass naked, the cooking pot obscuring Faendal's bits...
 "Are you lost" says Lydia!
Erm... yes... I think I'll just go back outside now!
No rhyme or reason for this happening. It just did. All part of what makes Skyrim great!
For those playing on a games console & wondering what the hell a "script" is, or an "AI package"... it's part of the program (game). You can't see it or edit it on a console though. You need a PC to do that!

A: In oblivion people would steal from containers you didnt own. The only way in THAT game to store something safely was to buy a house and store it in your house. I dont play skyrim so i cant say for sure, but being that its the sequel, id say its a safe bet that they work similarly.

A: Okay these guys have some of the info right. What they lack is a understanding of game engines and programming, second comment has a grasp, but clearly doesn't work with game engines. I have played Skyrim on ps3 for a few years now, pc has options for the game engine that are locked down in console version, I have no mods on my ps3. I have no downloaded content for Skyrim either, it is a pure original copy besides updates. So with that said I will explain the truth.
First off yes if you own the house no one can steal from you, that is not to say you won't find people in your home every now and then, but your stuff is safe no matter the container. As long as you own the house.
Second random places in the world to store your stuff is also correct in the statement "it is not safe", but no you can fix the re-spawning of object with in containers, when you find a location that did not have humanoid NPC's in it. (As even after you clear some of these locations the humanoid NPC's can re-populate the cave/whatever location, if you don't return to the location every day or so they will. Also it's not all of them at once, it is a slow gradual move in of thugs.)
Empty the chest of everything it had in it to start with, once all the objects have been removed the game engine will no longer re-spawning the objects into that said container, barrels and sacks will re-spawning no matter what. The you can store what ever you like in the said container and not have it go missing.
As for the thieves in Riverfront, well they like the Riverwood trader. I have found NPC's from the thieves guild in his store casing the place waiting for night time. He runs back and forth from upstairs to the opposite side on the ground floor during the day time. He also every now and then tries to pull out a dagger, when he dose the store owner says something about not doing that and the theft puts the dagger away. The thieves will steal from Riverfront trader, making that your home is not a great idea, buy a home in any of the holds (big cities) and you stuff is always guaranteed to be safe.
The reason they do steal from the Riverwood is because they are "scripted" (as one put it) to steal from him that is how the golden claw goes missing in the first place. Skyrim is built as if it is a virtual world the NPC's are programmed in such a way to respond to the environment (to some degree) this has been a president for all versions of elder scrolls since Morrowind.. That is not to say they will have random events, they are not AI but they do react to other NPC's and to the player. Random thieves in the field who try to rob you are a perfect example of this, if you have high stealth (best done with skill stealth at 100+) watch a theft when in stealth and fallow him, pick his pocket after he picks someone else's pocket you will see he did intact take a item from the other NPC's


Sometimes a case needs a different perspective, a neutral perspective. Someone who can see the gray between the white and black positions of the parties. Mediation is a powerful tool in helping the parties reach a resolution of their issues on their terms. Mediation encourages creative problem solving. The process is private, less costly than litigation, and tends to reduce the adversarial nature of court proceedings.
Within over twenty years of litigation experience, Derrick Wilson is available to serve as a civil or family mediator. Derrick Wilson is a registered Civil and Family Mediator in the state of Indiana and has successfully mediated dozens of cases. We look forward with helping you solve your legal issues with compassion, competence, and creativity.