text:
\section{Introduction}

The Grassmannian is the set of $k$-dimensional hyperplanes in $\mathbb{C}^n$. Lascoux and Sch\"utzenberger~\cite{LS82,LS83} introduced Grothendieck polynomials to represent the K-theory ring of the Grassmannian. In particular, they correspond to the K-theory classes of structure sheaves of Schubert varieties, and so they are indexed by permutations in $S_n$. By taking the stable limit of $n \to \infty$, Fomin and Kirillov~\cite{FK94,FK96} initiated the study of stable Grothendieck functions. The subset of stable Grothendieck functions corresponding to Grassmannian permutations can be indexed by partitions and are called symmetric Grothendieck polynomials. They are Schur positive and form a linearly independent set in the space of symmetric functions of any degree. In particular, the lowest degree term of a symmetric Grothendieck polynomial is the Schur polynomial indexed by the same partition.

For these reasons, it is natural to ask (1) how the involution, $\omega$, that interchanges elementary and homogeneous symmetric functions, affects symmetric Grothendieck polynomials, and (2) whether there is a set of polynomials orthonormal (under the Hall inner product: $\langle s_{\lambda},s_{\mu}\rangle = \delta_{\lambda,\mu}$) to the symmetric Grothendieck polynomials. These considerations led to the establishment of the four following versions of the symmetric Grothendieck polynomials and to the invention of the following combinatorial models to study them.

\begin{enumerate}
\item[(1A)] Symmetric Grothendieck polynomials using set-valued tableaux,
\item [(1B)] Weak symmetric Grothendieck polynomials using multiset-valued tableaux,
\item [(2A)] Dual symmetric Grothendieck polynomials using reverse plane partitions, and
\item [(2B)] Dual weak symmetric Grothendieck polynomials using valued-set tableaux.
\end{enumerate}

As the reader may guess from the names, (1A) goes to (1B) under $\omega$ and (2A) goes to (2B) under $\omega$ while (1A) is dual under $\langle,\rangle$ to (2A) and (1B) is dual under $\langle,\rangle$ to (2B). In other words we have the following diagram:

\begin{center}
$\begin{CD}
1A @>\omega>> 1B\\
@VV\langle,\rangle V @VV\langle,\rangle V\\
2A @>\omega>> 2B
\end{CD}$
\end{center}

Crystal analysis of these objects appearing in \cite{GAL17}, \cite{MPS18}, and \cite{HS20} all point toward a meaningful refinement of these polynomials using an additional parameter (in this paper $\mathbf{z}$). (For more on this, please see definition 3.2, Theorem 3.3, and remark 3.9 of \cite{CP19}.) Each of these refinements appears naturally in the underlying combinatorial object and denoting the new refined versions as \textbf{1A}, \textbf{1B}, \textbf{2A}, and \textbf{2B} we still have:

\begin{center}
$\begin{CD}
\mathbf{1A} @>\omega>> \mathbf{1B}\\
@VV\langle,\rangle V @VV\langle,\rangle V\\
\mathbf{2A} @>\omega>> \mathbf{2B}
\end{CD}$
\end{center}

As of yet, there has been little attempt to explain all these arrows in a single combinatorial framework. In order to do this, we begin by realizing each of the four combinatorial models as a particular instance of a certain type of tableau. Then we use a standard RSK argument and a standard jeu de taquin argument to find out what the fundamental combinatorial facts we need to prove are. As it turns out, a single combinatorial fact (main fact \ref{mf1}) explains both horizontal arrows, and another fact (main fact \ref{mf2}) explains both vertical arrows.

\subsection{Primed Tableaux}

\begin{definition}
Consider the alphabet $\{1'<1<2'<2<3'<3,\ldots\}$. An \emph{overfull tableau} of shape $\lambda$, or an element of $OT(\lambda)$ is a filling of a Young diagram of shape $\lambda$ such that
\begin{itemize}
\item Each box is nonempty and contains a set from $\{1',2',\ldots\}$ and a multiset from $\{1,2,\ldots\}$.
\item Suppose there is an $a$ in box $A$ and a $b$ in box $B$. Suppose box $B$ lies immediately to the right of box $B$. Then $a<b$ or else $a=b$ and both are unprimed numbers.
\item Suppose there is an $a$ in box $A$ and a $c$ in box $C$. Suppose box $C$ lies immediately below box $A$. Then $a<c$ or else $a=c$ and both are primed numbers.
\end{itemize}

An \emph{underfull tableau} of shape $\lambda$, or an element of $UT(\lambda)$ is a filling of a Young diagram of shape $\lambda$ such that
\begin{itemize}
\item Each box is empty, contains one number from $\{1',2',\ldots\}$, or, one number from $\{1,2,\ldots\}$., but no box in the leftmost column is empty.
\item Suppose there is an $a$ in box $A$ and a $b$ in box $B$. Suppose box $B$ lies to the right of box $A$ in the same row and is the leftmost such box that is nonempty. Then $a<b$ or else $a=b$ and both are unprimed numbers.
\item Suppose there is an $a$ in box $A$ and a $c$ in box $C$. Suppose box $C$ lies in the row below box $A$ and weakly to its left, and, is the rightmost such box that is nonempty. Then $a<c$ or else $a=c$ and both are primed numbers.
\end{itemize}

\end{definition}
The \emph{ left weight} of an $OT$ or a $UT$ is the vector whose $i^{th}$ coordinate records the number of times $i$ appears in the tableau. The \emph{right weight}of an $OT$ or a $UT$ is the vector whose $i^{th}$ coordinate records the number of times $i'$ appears in the tableau. The \emph{overweight} of an $OT$ of shape $\lambda$ is the vector whose $i^{th}$ coordinate records the difference between the number of entries in column $i$ and the number of boxes in column $i$. The \emph{underweight} of a $UT$ of shape $\lambda$ is the vector whose $i^{th}$ coordinate records the difference between the number of boxes in column $(i+1)$ and the number of entries in column $(i+1)$. (By convention, either of these differences is taken to be $0$ if the specified column is not part of the tableau.) A tableau which is both an overfull tableau and an underfull tableau (i.e., has exactly one entry per box) is called a primed tableau and the set of all such tableaux of shape $\lambda$ is denoted by $PT(\lambda)$.

\begin{definition}

Let $\lambda$ be a partition and let $\mathbf{x}=(x_1,x_2,\ldots)$, $\mathbf{y}=(y_1,y_2,\ldots)$, and $\mathbf{z}=(z_1,z_2,\ldots)$ be infinite sets of indeterminants. We define polynomials:
\begin{eqnarray*}
\mathfrak{G}_{\lambda}(\mathbf{x},\mathbf{y},\mathbf{z})&=&\sum_{T \in OT(\lambda)} \mathbf{x}^{\ell w(T)} \mathbf{y}^{rw(T)}\mathbf{z}^{O(T)}(-1)^{|O(T)|}\\
\mathfrak{G}^*_{\lambda}(\mathbf{x},\mathbf{y},\mathbf{z})&=&\sum_{T \in UT(\lambda)} \mathbf{x}^{\ell w(T)} \mathbf{y}^{rw(T)}\mathbf{z}^{U(T)}
\end{eqnarray*}
Here, $\ell w(T)$ is the left weight of $T$, $rw(T)$ is the right weight of $T$, $O(T)$ is the overweight of $T$, and $U(T)$ is the underweight of $T$.
\end{definition}

\begin{example} An overfull and an underfull tableau.
\begin{multicols}{2}
$P=\Yboxdim{24pt}
\young({{1'11}}{{12'}}{{23'}},{{2'}}{{2}}{{3'33}},{{2'3'}}{{3}})$\\
\begin{itemize}
\item $P \in OT(3,3,2)$
\item $\ell w(P)=(2,1,3)$
\item $rw(P)=(1,3,3)$
\item $O(P)=(3,1,3)$
\end{itemize}
$Q=\Yboxdim{18pt} \young({{1'}}{{}}{{1}}{{}}{{1}},{{1}}{{2'}}{{}}{{2}}{{}},{{2'}}{{}}{{3}},{{3}})$\\
\begin{itemize}
\item $Q \in UT(4,4,3,1)$
\item $\ell w(Q)=(3,1,2)$
\item $rw(Q)=(1,2,0)$
\item $U(Q)=(2,1,1,1)$
\end{itemize}
\end{multicols}

\end{example}

\begin{definition}
Let $\lambda$ be a partition with conjugate, $\lambda'$. For the partition $\lambda$ the
\begin{enumerate}
\item [(\textbf{1A})] refined symmetric Grothendieck polynomial is $\mathfrak{G}_{\lambda'}(\mathbf{0},\mathbf{x},\mathbf{z})$.
\item [(\textbf{1B})] refined weak symmetric Grothendieck polynomial is $\mathfrak{G}_{\lambda}(\mathbf{x},\mathbf{0},\mathbf{z})$.
\item [(\textbf{2A})] refined dual symmetric Grothendieck polynomial is $\mathfrak{G}^*_{\lambda'}(\mathbf{0},\mathbf{x},\mathbf{z})$.
\item [(\textbf{2B})] refined dual weak symmetric Grothendieck polynomial is $\mathfrak{G}^*_{\lambda}(\mathbf{x},\mathbf{0},\mathbf{z})$.
\end{enumerate}
The nonrefined versions of these polynomials are obtained by setting $\mathbf{z}=\mathbf{1}$.
\end{definition}
It is not hard to see that these definitions coincide with the combinatorial definitions of these polynomials given elsewhere.
For the nonrefined versions of each of these four polynomials see, respectively, Theorem 3.1 of \cite{Buch02}, definition 9.20 of \cite{Lenart00}, corollary 2 of \cite{GAL17}, and definition 9.23 of \cite{Lenart00}. To see an example of the refined versions see definition 3.2 and Theorem 3.3 of \cite{CP19}.

\begin{definition}

Let $\mu \subseteq \lambda$ be partitions with an equal number of rows. An \emph{over flagged tableau} of shape $\lambda/\mu$, or an element of $OFT(\lambda/\mu)$, is a filling of a Young diagram of shape $\lambda/\mu$ using the alphabet $1<2<\cdots$ such that:
\begin{itemize}
\item Each box in row $i$ of $\lambda/\mu$ contains one element from $\{1,2,\ldots,\mu_i\}$.
\item Suppose box $A$ lies immediately to the left of box $B$. Suppose there is an $a$ in $A$ and a $b$ in $B$. Then $a\geq b$.
\item Suppose box $A$ lies immediately to the above of box $C$. Suppose there is an $a$ in $A$ and a $c$ in $C$. Then $a>c$.
\end{itemize}
The weight, $wt(P)$, of an $OFT$, $P$, is the vector whose $i^{th}$ coordinate records the number of $i$s in the tableau.

Let $\mu \subseteq \lambda$ be partitions with an equal number of rows. An \emph{under flagged tableau} of shape $\lambda/\mu$, or an element of $UFT(\lambda/\mu)$ is a filling of a Young diagram of shape $\lambda/\mu$ using the alphabet $1' <2'<\cdots$ such that:
\begin{itemize}
\item Each box in row $i$ of $\lambda/\mu$ contains one element from $\{1',\ldots,(\lambda_i-1)'\}$.
\item Suppose box $A$ lies immediately to the left of box $B$. Suppose there is an $a$ in $A$ and a $b$ in $B$. Then $a< b$.
\item Suppose box $A$ lies immediately to the above of box $C$. Suppose there is an $a$ in $A$ and a $c$ in $C$. Then $a\leq c$.
\end{itemize}
The weight, $wt(P)$, of a $UFT$, $P$ is the vector whose $i^{th}$ coordinate records the number of $i'$s in the tableau.
\end{definition}

\begin{example} An over flagged and an under flagged tableau.
\begin{multicols}{2}
$P=\Yboxdim{18pt}
\young(****42,***321,**221,*111)$\\
\begin{itemize}
\item $P \in OFT((6,6,5,4)/(4,3,2,1))$
\item $wt(P)=(5,3,1,1)$
\end{itemize}
$Q=\Yboxdim{18pt}
\young(****{{1'}}{{5'}},***{{2'}}{{3'}}{{5'}},**{{1'}}{{2'}}{{4'}},*{{1'}}{{2'}}{{3'}})$\\
\begin{itemize}
\item $Q \in UFT((6,6,5,4)/(4,3,2,1))$
\item $wt(Q)=(3,3,2,1,2)$
\end{itemize}
\end{multicols}

\end{example}
If $\lambda \supseteq \mu$ have a different number of rows, set $OFT(\lambda/\mu)=\emptyset=UFT(\lambda/\mu)$.

\begin{lemma}\label{rskjdt} There are bijections between the following sets:
\begin{enumerate}
\item $OT(\mu)$ and $\{(P,Q): \exists \lambda \supseteq \mu: P \in PT(\lambda),Q \in OFT(\lambda/\mu)\}$.
\item $UT(\lambda)$ and $\{(P,Q): \exists \mu \subseteq \lambda: P \in PT(\mu),Q \in UFT(\lambda/\mu)\}$.
\end{enumerate}
In case (1), if $T \rightarrow (P,Q)$ then $\ell w(T)=\ell w(P)$, $rw(T)=rw(P)$, and $O(T)=wt(Q)$.\\
In case (2), if $T \rightarrow (P,Q)$ then $\ell w(T)=\ell w(P)$, $rw(T)=rw(P)$, and $U(T)=wt(Q)$.
\end{lemma}

\begin{proof}
We construct each bijection. We will use RSK column insertion in the first and jeu de taquin in the second. The proper versions of these algorithms to use are those that are obtained from the standard versions (for alphabets of distinct letters) by beginning with the order $1'<1<2'<2<\cdots$ and standardizing the unprimed entries of a tableau left to right and the primed entries of a tableau top to bottom.
\begin{enumerate}
\item Start with $T \in OT(\mu)$ and construct $(P,Q)$ as follows. Beginning with the rightmost column, that is column $\mu_1$, and working to the leftmost column, that is, column $1$, do as follows. From each box in the current column, say column $i$, remove all but the smallest entry. Now, in weakly decreasing order, insert the removed entries to the tableau formed by the columns $i+1,i+2,\ldots$ using RSK insertion. Let $P$ be the resulting tableau and suppose it has shape $\lambda$. Now construct a tableau $Q$ of shape $\lambda/\mu$ by placing an $i$ in each box that corresponds to the position of a box appended during the RSK insertions that occurred after removing entries from column $i$.
\item Start with $T \in UT(\lambda)$ and construct $(P,Q)$ as follows. Beginning with the rightmost column, that is column $\lambda_1$, and working to the second to leftmost column, that is, column $2$, do as follows. Starting with the lowest and working to the highest, do the following for each empty box, $b$, in column $i$. Consider the tableau, $R$, whose upper left corner box is $b$. $R$ has exactly one empty box which is box $b$. Apply jdt into box $b$ and continue until an outer corner box is removed from $R$. After this has been done for each box $b$ for each column $i$ from $\lambda_1$ to $2$ define the result to be $P$ and denote its shape by $\mu$. Now construct a tableau $Q$ of shape $\lambda/\mu$ by placing a $(i-1)'$ in each box that corresponds to the position of a box removed after a jdt move into a box in column $i$.
\end{enumerate}
The $P$-tableaux constructed are valid primed tableaux by the properties of RSK and jdt. The $Q$-tableaux constructed are valid column strict and row strict tableaux respectively, also by properties of RSK and jdt. On the other hand, the flag conditions for an $OT$ or a $UT$ are ensured directly from the construction. Moreover, the weight conditions $\ell w(T)=\ell w(P)$, $rw(T)=rw(P)$ are immediate since RSK and jdt don't affect the set of entries appearing in a tableau, and the weight conditions $O(T)=wt(Q)$ and $U(T)=wt(Q)$ follow by construction. Finally, it is not difficult to see how to invert the maps using the fact that RSK and jdt are themselves invertible.

\end{proof}

\begin{example}
An example of bijection $(1)$ of Lemma \ref{rskjdt}. In each step the newly colored entry is to be RSK inserted into the column to its right. The final colored tableau is $P$ and $Q$ is shown directly below it.
\begin{eqnarray*}
\Yboxdim{23pt}
\young({{1'}}{{2'3'}},{{12'3'}}{{3\textcolor{red}{3}}},{{3'4'4}}) \longrightarrow
\young({{1'}}{{2'\textcolor{blue}{3'}}}{{\textcolor{red}{3}}},{{12'3'}}{{3}},{{3'4'4}}) \longrightarrow
\young({{1'}}{{2'}}{{\textcolor{blue}{3'}}}{{\textcolor{red}{3}}},{{12'3'}}{{3}},{{3'4'\textcolor{green}{4}}}) \longrightarrow
\young({{1'}}{{2'}}{{\textcolor{blue}{3'}}}{{\textcolor{red}{3}}},{{12'3'}}{{3}},{{3'\textcolor{brown}{4'}}}{{\textcolor{green}{4}}})\\
\Yboxdim{23pt}
\longrightarrow \young({{1'}}{{2'}}{{\textcolor{blue}{3'}}}{{\textcolor{red}{3}}},{{12'\textcolor{orange}{3'}}}{{3}}{{\textcolor{green}{4}}},{{3'}}{{\textcolor{brown}{4'}}})
\longrightarrow \young({{1'}}{{2'}}{{\textcolor{blue}{3'}}}{{\textcolor{red}{3}}},{{1\textcolor{gray}{2'}}}{{\textcolor{orange}{3'}}}{{3}}{{\textcolor{green}{4}}},{{3'}}{{\textcolor{brown}{4'}}})
\longrightarrow \young({{1'}}{{2'}}{{\textcolor{blue}{3'}}}{{3}}{{\textcolor{red}{3}}},{{1}}{{\textcolor{gray}{2'}}}{{\textcolor{orange}{3'}}}{{\textcolor{green}{4}}},{{3'}}{{\textcolor{brown}{4'}}})\\
\Yboxdim{23pt} \young(**221,**11,*1)
\end{eqnarray*}
\end{example}

\begin{example}
An example of bijection $(2)$ of Lemma \ref{rskjdt}. In each step the jeu de taquin algorithm is to be applied into the box marked by the $\textcolor{red}{\times}$. The final tableau is $P$ and $Q$ is shown directly below it.

\begin{eqnarray*}
\Yboxdim{12pt}
\young({{1'}}{{}}{{}}{{}}{{3'}},{{2'}}{{}}{{3'}}{{}}{{\textcolor{red}{\times}}},{{2}}{{3}}{{}}{{4}}{{4}},{{4}}{{}}{{4}})\longrightarrow
\young({{1'}}{{}}{{}}{{}}{{3'}},{{2'}}{{}}{{3'}}{{\textcolor{red}{\times}}}{{4}},{{2}}{{3}}{{}}{{4}},{{4}}{{}}{{4}})\longrightarrow
\young({{1'}}{{}}{{}}{{\textcolor{red}{\times}}}{{3'}},{{2'}}{{}}{{3'}}{{4}}{{4}},{{2}}{{3}}{{}},{{4}}{{}}{{4}})\longrightarrow
\young({{1'}}{{}}{{}}{{3'}}{{4}},{{2'}}{{}}{{3'}}{{4}},{{2}}{{3}}{{\textcolor{red}{\times}}},{{4}}{{}}{{4}})\longrightarrow\\
\Yboxdim{12pt}
\young({{1'}}{{}}{{\textcolor{red}{\times}}}{{3'}}{{4}},{{2'}}{{}}{{3'}}{{4}},{{2}}{{3}}{{4}},{{4}}{{}})\rightarrow
\young({{1'}}{{}}{{3'}}{{4}}{{4}},{{2'}}{{}}{{3'}},{{2}}{{3}}{{4}},{{4}}{{\textcolor{red}{\times}}})\rightarrow
\young({{1'}}{{}}{{3'}}{{4}}{{4}},{{2'}}{{\textcolor{red}{\times}}}{{3'}},{{2}}{{3}}{{4}},{{4}})\rightarrow
\young({{1'}}{{\textcolor{red}{\times}}}{{3'}}{{4}}{{4}},{{2'}}{{3'}}{{4}},{{2}}{{3}},{{4}}) \rightarrow
\young({{1'}}{{3'}}{{4}}{{4}}{{4}},{{2'}}{{3'}},{{2}}{{3}},{{4}})\\
\Yboxdim{12pt}
\young(*****,**{{1'}}{{2'}}{{3'}},**{{1'}}{{3'}}{{4'}},*{{1'}}{{2'}})
\end{eqnarray*}
\end{example}

The following two main facts are the main results of this paper. These facts along with the results so far stated give a complete combinatorial understanding of two types of duality appearing in Grothendieck polynomials. These facts will be proven in the next section.

\begin{mainfact}\label{mf1}
Let $\omega_{\mathbf{x}}$ denote the involution on functions symmetric in $\mathbf{x}$ over the ring $\mathbb{Z}[\mathbf{y},\mathbf{z}]$ defined by $\omega_{\mathbf{x}}(s_{\lambda}(\mathbf{x}))=s_{\lambda'}(\mathbf{x})$. Similarly, let $\omega_{\mathbf{y}}$ denote the involution on functions symmetric in $\mathbf{y}$ over the ring $\mathbb{Z}[\mathbf{x},\mathbf{z}]$ defined by $\omega_{\mathbf{y}}(s_{\lambda}(\mathbf{y}))=s_{\lambda'}(\mathbf{y})$. Then we have that:
\begin{eqnarray*}
\omega_{\mathbf{x}}\omega_{\mathbf{y}}\left(\sum_{P \in PT(\lambda)} \mathbf{x}^{\ell w(P)} \mathbf{y}^{rw(P)}\right)=\sum_{P \in PT(\lambda)} \mathbf{y}^{\ell w(P)} \mathbf{x}^{rw(P)}
\end{eqnarray*}
\end{mainfact}

\begin{corollary}\label{duo1} We have:
\begin{align*}
\omega_{\mathbf{x}}\omega_{\mathbf{y}}\mathfrak{G}_{\mu}(\mathbf{x},\mathbf{y},\mathbf{z})=\mathfrak{G}_{\mu}(\mathbf{y},\mathbf{x},\mathbf{z})\\
\omega_{\mathbf{x}}\omega_{\mathbf{y}}\mathfrak{G}^*_{\lambda}(\mathbf{x},\mathbf{y},\mathbf{z})=\mathfrak{G}^*_{\lambda}(\mathbf{y},\mathbf{x},\mathbf{z})\\
\end{align*}
\end{corollary}

\begin{proof}
Using Lemma \ref{rskjdt} we may write:
\begin{align*}
&\mathfrak{G}_{\mu}(\mathbf{x},\mathbf{y},\mathbf{z})=\sum_{\lambda \supseteq \mu}& (-1)^{|\lambda|-|\mu|}&\sum_{Q \in OFT(\lambda/\mu)}\mathbf{z}^{O(Q)}\sum_{P \in PT(\lambda)} \mathbf{x}^{\ell w(P)} \mathbf{y}^{rw(P)}&\\
&\mathfrak{G}^*_{\lambda}(\mathbf{x},\mathbf{y},\mathbf{z})=\sum_{\mu \subseteq \lambda} & & \sum_{Q \in UFT(\lambda/\mu)}\mathbf{z}^{U(Q)}\sum_{P \in PT(\lambda)} \mathbf{x}^{\ell w(P)} \mathbf{y}^{rw(P)}&\\
\end{align*}
The corollary now follows from main fact \ref{mf1} and the linearity of $\omega_{\mathbf{x}}$ and $\omega_{\mathbf{y}}$.
\end{proof}

\begin{remark}
Setting $\mathbf{y}=\mathbf{0}$ in the first equation of corollary \ref{duo1} we see that the refined weak symmetric Grothendieck polynomial maps under $\omega_{\mathbf{x}}$ to the refined symmetric Grothendieck polynomial of conjugate shape. Setting $\mathbf{y}=\mathbf{0}$ in the second equation of corollary \ref{duo1} we see that the refined dual weak symmetric Grothendieck polynomial maps under $\omega_{\mathbf{x}}$ to the refined dual symmetric Grothendieck polynomial of conjugate shape. In other words this shows $\textbf{1A} \xrightarrow{\omega} \textbf{1B}$ and $\textbf{2A} \xrightarrow{\omega} \textbf{2B}$.
\end{remark}

\begin{mainfact}\label{mf2}
Let $\mu \subset \lambda$ be partitions with the same number of rows. Then:
\begin{eqnarray*}
\sum_{\mu \subseteq \rho \subseteq \lambda} \sum_{(P,Q) \in T_{\rho}}(-1)^{|\rho|-|\mu|}\mathbf{z}^{wt(P)}\mathbf{z}^{wt(Q)}=0
\end{eqnarray*}
where $T_{\rho}$ is the set of all pairs $(P,Q)$ with $P \in OFT(\rho/\mu)$ and $Q \in UFT(\lambda/\rho)$.
\end{mainfact}

\begin{corollary}\label{duo2}
Let $\langle,\rangle$ denote the bilinear form on functions symmetric in $\mathbf{x}$ over the ring $\mathbb{Z}[\mathbf{z}]$ defined by $\langle s_{\mu}(\mathbf{x}),s_{\lambda}(\mathbf{x})\rangle=\delta_{\mu,\lambda}$ and extended $\mathbb{Z}[\mathbf{z}]$-bilinearly. Then we have:
\begin{eqnarray*}
\langle \mathfrak{G}_{\mu}(\mathbf{x},\mathbf{0},\mathbf{z}), \mathfrak{G}^*_{\lambda}(\mathbf{x},\mathbf{0},\mathbf{z}) \rangle = \delta_{\mu,\lambda}\\
\langle \mathfrak{G}_{\mu}(\mathbf{0},\mathbf{x},\mathbf{z}), \mathfrak{G}^*_{\lambda}(\mathbf{0},\mathbf{x},\mathbf{z}) \rangle = \delta_{\mu,\lambda}\\
\end{eqnarray*}

\end{corollary}

\begin{proof}

Since a primed tableau with right weight (resp. leftweight) of $\mathbf{0}$ is just a semistandard Young tableau (resp. conjugated semistandard Young tableau) it follows from the formulas in the proof of corollary \ref{duo1} that $\langle \mathfrak{G}_{\mu}(\mathbf{x},\mathbf{0},\mathbf{z}), \mathfrak{G}^*_{\lambda}(\mathbf{x},\mathbf{0},\mathbf{z}) \rangle$ is equal to:
\begin{align*}
\langle \left( \sum_{\rho \supseteq \mu} (-1)^{|\rho|-|\mu|}\sum_{P \in OFT(\rho/\mu)}\mathbf{z}^{O(P)}s_{\rho}(\mathbf{x})\right) , \left(\sum_{\sigma \subseteq \lambda} \sum_{Q \in UFT(\lambda/\sigma)}\mathbf{z}^{U(Q)}s_{\sigma}(\mathbf{x})\right)\rangle\\
\end{align*}
and that $\langle \mathfrak{G}_{\mu}(\mathbf{0},\mathbf{x},\mathbf{z}), \mathfrak{G}^*_{\lambda}(\mathbf{0},\mathbf{x},\mathbf{z}) \rangle$ is equal to:
\begin{align*}
\langle \left( \sum_{\rho \supseteq \mu} (-1)^{|\rho|-|\mu|}\sum_{P \in OFT(\rho/\mu)}\mathbf{z}^{O(P)}s_{\rho'}(\mathbf{x})\right) , \left(\sum_{\sigma \subseteq \lambda} \sum_{Q \in UFT(\lambda/\sigma)}\mathbf{z}^{U(Q)}s_{\sigma'}(\mathbf{x})\right)\rangle\\
\end{align*}
If $\mu=\lambda$ then it is clear there is exactly one nonzero term in the expansion of these inner products which occurs when $\mu=\rho=\sigma$ and that this term has coefficient $(-1)^0 \mathbf{z}^{\mathbf{0}}=1$. If $\mu \not\subseteq \lambda$ or $\mu \subset \lambda$ have different numbers of rows then it is clear that all terms in the expansion of the inner products must be $0$. Finally, if $\mu \subset \lambda$ then it follows from main fact \ref{mf2} that the inner products evaluate to $0$.
\end{proof}

\begin{remark}
Corollary \ref{duo2} says that under $\langle,\rangle$ the refined weak symmetric Grothendieck polynomial is dual to the refined dual weak symmetric Grothendieck polynomial and that the refined symmetric Grothendieck polynomial is dual to the refined dual symmetric Grothendieck polynomial. I.e., $\textbf{1A} \xrightarrow{\langle,\rangle} \textbf{2A}$ and $\textbf{1B} \xrightarrow{\langle,\rangle} \textbf{2B}$.
\end{remark}

\section{Proof of the Main Facts}

\begin{definition}
Let $\prec$ denote any total order on the set $\{1',1,2',2',\ldots\}$. We define a primed tableau with respect to $\prec$ of shape $\lambda/\mu$ to be a filling of the shape $\lambda/\mu$ such that:
\begin{itemize}
\item Each box of $\lambda/\mu$ contains exactly one of $\{1',1,2',2',\ldots\}$.
\item The entries in the rows of $\lambda/\mu$ weakly increase under $\prec$ left to right.
\item The entries in the columns of $\lambda/\mu$ weakly increase $\prec$ top to bottom.
\item There is at most one $i$ in each column for each $i$.
\item There is at most one $i'$ in each row for each $i$.
\end{itemize}
The set of such tableaux is denoted by $PT_{\prec}(\lambda/\mu)$. The left weight of a $PT_{\prec}$ is the vector whose $i^{th}$ coordinate records the number of times $i$ appears in the tableau. The right weight of a $PT_{\prec}$ is the vector whose $i^{th}$ coordinate records the number of times $i'$ appears in the tableau.
\end{definition}
Note that if $\prec$ is the order $1'\prec1\prec2'\prec2\prec \cdots$ then $PT_{\prec}(\lambda/\emptyset)=PT(\lambda)$.

\begin{lemma}\label{ordering}
The total order chosen in the definition above is irrelevant. In other words, given any two total orderings $\prec$ and $\vartriangleleft$ there is a left weight and right weight preserving bijection between $PT_{\prec}(\lambda/\mu)$ and $PT_{\vartriangleleft}(\lambda/\mu)$.
\end{lemma}

\begin{proof}
First we prove the lemma in the special case that $\prec$ coincides with $\vartriangleleft$ when restricted to $\{1,2,\ldots\}$ and when restricted to $\{1',2',\ldots\}$. To do this it suffices to suppose that there are some $i$ and $j$ (possibly equal) such that $i \prec j'$ and $j'\vartriangleleft i$ and all other pairs of letters have the same relationship in both orders. From this assumption it follows that in both orders there is no other letter between $i$ and $j'$.

If $T \in PT_{\prec}(\lambda/\mu)$ or $T \in PT_{\vartriangleleft}(\lambda/\mu)$, let $\mathcal{B}(T)$ denote all of the boxes of $\lambda/\mu$ that contain an $i$ or a $j'$. If $\lambda=(\lambda_1,\ldots,\lambda_{\ell})$ then label the the upper left to lower right diagonals of $\lambda$ by $\{D_{-\ell+1}, \ldots,D_{-1},D_0,D_1,\ldots, D_{\lambda_1-1}\}$. It is clear that either $\mathcal{B}(T) \cap D_s$ is a single box or is empty for each $s$. If $p$ is minimal and $q$ is maximal such that the adjacent diagonals $d_p,d_{p+1},\ldots,d_{q-1},d_q$ each have nonempty intersection with $\mathcal{B}(T)$ then we call $B=\{d_p \cap \mathcal{B}(T) , \ldots, d_q \cap \mathcal{B}(T)\}$ a connected component of $\mathcal{B}(T)$.

Define a map $\nearrow$ from $PT_{\prec}(\lambda/\mu)$ to $PT_{\vartriangleleft}(\lambda/\mu)$ as follows. Suppose that $T \in PT_{\prec}(\lambda/\mu)$. Perform the following to each connected component $B$ of $\mathcal{B}(T)$.
\begin{itemize}
\item Remove the entry that appears in the upper rightmost box of $B$. Record what you have removed.
\item Move every remaining $i$ one box to the right.
\item Move every remaining $j'$ one box up.
\item Fill the lower leftmost box of $B$ with the entry recorded in step one.
\end{itemize}

Next define a map $\swarrow$ from $PT_{\vartriangleleft}(\lambda/\mu)$ to $PT_{\prec}(\lambda/\mu)$ as follows. Suppose that $T \in PT_{\vartriangleleft}(\lambda)$. Perform the following to each connected component $B$ of $\mathcal{B}(T)$.
\begin{itemize}
\item Remove the entry that appears in the lower leftmost box of $B$. Record what you have removed.
\item Move every remaining $i$ one box to the left.
\item Move every remaining $j'$ one box down.
\item Fill the upper rightmost box of $B$ with the entry recorded in step one.
\end{itemize}

It is not difficult to check that the maps $\nearrow$ and $\swarrow$ are well-defined and preserve weights and are mutual inverses. This proves the special case. Thus, to prove the general case we may assume, for instance, that for all $i$ and all $j'$ we have $i \prec j'$ and $i \vartriangleleft j'$. But in this case a primed tableau may be thought of as a pair of tableaux composed of the tableau formed by the unprimed entries and the tableau formed by the primed entries. Thus we may write:
\begin{eqnarray*}
PT_{\prec}(\lambda/\mu)= \bigcup_{\mu \subseteq \rho \subseteq \lambda} SSYT_{\prec}(\rho/\mu) \times SSYT'_{\prec}(\lambda/\rho)
\end{eqnarray*}
where $SSYT_{\prec}(\rho/\mu)$ is the set of skew semistandard Young tableaux of shape $(\rho/\mu)$in the alphabet $\{1,2,\cdots\}$ under the order $\prec$ as restricted to $\{1,2,\cdots\}$ and $SSYT'_{\prec}(\lambda/\rho)$ is the set of conjugate skew semistandard Young tableaux of shape $(\lambda/\rho)$ in the alphabet $\{1',2',\cdots\}$ under the order $\prec$ as restricted to $\{1',2',\cdots\}$. But all skew Schur functions are symmetric so this implies that $\prec$ may be replaced with $\vartriangleleft$ in the right hand side of the equation above, which makes it equal to $PT_{\vartriangleleft}(\lambda/\mu)$ by the same logic with which we arrived at this equation.

\end{proof}

\begin{example}
Suppose that we are given the following orderings:
\begin{eqnarray*}
1' \prec 1 \prec 2 \prec 3 \prec 2' \prec 4 \prec 3' \prec 4'\\
1' \vartriangleleft 1 \vartriangleleft 2 \vartriangleleft 2' \vartriangleleft 3 \vartriangleleft 4 \vartriangleleft 3' \vartriangleleft 4'
\end{eqnarray*}
Then under the map $\nearrow$ with $i=3$ and $j=2$ the tableau $S \in PT_{\prec}(\lambda/\mu)$ on the left below is sent to the tableau $T \in PT_{\vartriangleleft}(\lambda/\mu)$ on the right below. Note that $\mathcal{B}(S)$ is composed of $3$ connected components of sizes $1$, $7$, and $3$.
\begin{eqnarray*}
S=\young(::{{1'}}122{{\textcolor{red}{3}}}{{\textcolor{red}{3}}}{{\textcolor{red}{2'}}},{{1'}}11{{\textcolor{red}{3}}}{{\textcolor{red}{3}}}{{\textcolor{red}{3}}}{{3'}}{{4'}},{{1'}}22{{\textcolor{red}{2'}}}44{{3'}},1{{\textcolor{red}{3}}}{{\textcolor{red}{3}}}{{\textcolor{red}{2'}}}{{3'}},24{{3'}},{{\textcolor{red}{3}}}{{3'}})
\nearrow \,\,\,\,\,\,\,\,\,\,\,\,
\young(::{{1'}}122{{\textcolor{red}{2'}}}{{\textcolor{red}{3}}}{{\textcolor{red}{3}}},{{1'}}11{{\textcolor{red}{2'}}}{{\textcolor{red}{3}}}{{\textcolor{red}{3}}}{{3'}}{{4'}},{{1'}}22{{\textcolor{red}{2'}}}44{{3'}},1{{\textcolor{red}{3}}}{{\textcolor{red}{3}}}{{\textcolor{red}{3}}}{{3'}},24{{3'}},{{\textcolor{red}{3}}}{{3'}})=T
\end{eqnarray*}
\end{example}

\begin{proof}[Proof of Main Fact \ref{mf1}]
Consider the orders:
\begin{eqnarray*}
1'\prec 1 \prec 2' \prec 2 \prec \cdots\\
1\vartriangleleft 2 \vartriangleleft \cdots 1' \vartriangleleft 2' \vartriangleleft \cdots\\
1' \blacktriangleleft 2' \blacktriangleleft \cdots 1 \blacktriangleleft 2 \blacktriangleleft \cdots
\end{eqnarray*}

Since an element of $ PT_{\vartriangleleft}(\lambda/\emptyset)$ can be thought as a semistandard Young tableau of some shape $\mu \subseteq \lambda$ along with a conjugate semistadard Young tableau (with its entries primed) of shape $\lambda/\mu$, we see that:
\begin{eqnarray*}
\sum_{P \in PT_{\vartriangleleft}(\lambda/\emptyset)} \mathbf{x}^{\ell w(P)} \mathbf{y}^{rw(P)}=\sum_{\mu \subseteq \lambda} s_{\mu}(\mathbf{x}) s_{\lambda'/\mu'}(\mathbf{y})
\end{eqnarray*}

On the other hand, an element $PT_{\blacktriangleleft}(\lambda/\emptyset)$ can be thought as a conjugate semistandard Young tableau (with its entries primed) of some shape $\mu \subseteq \lambda$ along with a semistandard Young tableau of shape $\lambda/\mu$, so that:
\begin{eqnarray*}
\sum_{P \in PT_{\blacktriangleleft}(\lambda/\emptyset)} \mathbf{y}^{\ell w(P)} \mathbf{x}^{rw(P)}=\sum_{\mu \subseteq \lambda} s_{\mu'}(\mathbf{x}) s_{\lambda/\mu}(\mathbf{y})
\end{eqnarray*}

Since the right hand side of the equation above is obtained from the right hand side of the equation before it by applying $\omega_{\mathbf{x}}\omega_{\mathbf{y}}$ the same is true of the left hand sides. Moreover by Lemma \ref{ordering} both $PT_{\vartriangleleft}(\lambda/\emptyset)$ and $PT_{\blacktriangleleft}(\lambda/\emptyset)$ may be replaced by $PT_{\prec}(\lambda/\emptyset)$ which is turn equivalent to $PT(\lambda)$. This proves that:
\begin{eqnarray*}
\omega_{\mathbf{x}}\omega_{\mathbf{y}}\left(\sum_{P \in PT(\lambda)} \mathbf{x}^{\ell w(P)} \mathbf{y}^{rw(P)}\right)=\sum_{P \in PT(\lambda)} \mathbf{y}^{\ell w(P)} \mathbf{x}^{rw(P)}
\end{eqnarray*}

\end{proof}

\begin{definition}
Fix the order $\cdots \prec 2 \prec 1 \prec 1' \prec 2' \prec \cdots$. Define a primed flagged tableau of shape $\lambda/\mu$ to be a tableau, $P$, that is an element of $PT_{\prec}(\lambda/\mu)$ such that:
\begin{itemize}
\item If row $i$ of $P$ contains a $j$ then $j \leq \mu_i$
\item If row $i$ of $P$ contains a $j'$ then $j < \lambda_i$.
\end{itemize}
Let $PFT(\lambda/\mu)$ denote the set of all primed flagged tableaux of shape $\lambda/\mu$. We say a ``row is $\prec$" if the entries are weakly increasing left ro right under $\prec$ and there are no repeated primed entries. We say a ``column is $\prec$" if the entries are weakly increasing top to bottom under $\prec$ and there are no repeated unprimed entries. The condition concerning the nonprimed entries of row $i$ is called the nonprimed row $i$ flag condition. The condition concerning the primed entries of row $i$ is called the primed row $i$ flag condition. In other words, $P \in PFT(\lambda/\mu)$ if and only all its rows and columns are $\prec$ and it satisfies the nonprimed and primed row $i$ flag conditions for each $i$.

\end{definition}

\begin{lemma}\label{z}
Let $\mu \subset \lambda$ be partitions with the same number of rows.
\begin{eqnarray*}
\sum_{P \in PFT_{\prec}(\lambda/\mu)} (-1)^{\#(P)}\mathbf{z}^{\ell w(P)}\mathbf{z}^{rw(P)}=0
\end{eqnarray*}
Where $\#(P)$ is the number of nonprimed entries in $P$, $\ell w(P)$ is the left weight of $P$, and $rw(P)$ is the right weight of $P$.
\end{lemma}
\begin{proof}
We complete the proof by constructing a sign changing involution on $ PFT(\lambda/\mu)$ which preserves the value of $\ell w(P)+rw(P)$ and where the sign is the value of $(-1)^{\#(P)}$. The involution, $\iota$, is described as follows:

Let $P \in PFT(\lambda/\mu)$. Let $m$ be the smallest integer such that $P$ contains an $m$ or an $m'$. Now, let $i$ be minimal such that row $i$ of $P$ contains an $m$ or $m'$. Finally, let $j$ be maximal such that the box, $b$, of $P$, in row $i$ and column $j$ contains an $m$ or an $m'$. Define $\iota(P)$ to be the tableau obtained by:
\begin{itemize}
\item If $b$ contains an $m$, replace it with an $m'$.
\item If $b$ contains an $m'$, replace it with an $m$.
\end{itemize}

First suppose that box $b$ of $P$ contains an $m$. If $\lambda/\mu$ has a box to the right of $b$ then $P$ must have in this box an $s$ or an $s' $ for some $s>m$ by the minimality of $m$ and the maximality of $j$. But it cannot be an $s$ because row $i$ of $P$ is assumed to be $\prec$. Since $m' \prec s'$ it follows that row $i$ of $\iota(P)$ is $\prec$.
If $\lambda/\mu$ has a box below $b$ then $P$ must have in this box a $t$ or a $t' $ for some $t \geq m$ by the minimality of $m$. But it cannot be a $t$ because column $j$ of $P$ is assumed to be $\prec$. Since $m' \preceq t'$ it follows that column $j$ of $\iota(P)$ is $\prec$. Consequently, all rows and columns of $\iota(P)$ are $\prec$.
Next, we have $m \leq \mu_i$ because $P$ is assumed to satisfy the nonprimed row $i$ flag condition. But $\mu_i<\lambda_i$ since we know row $i$ of $\lambda/\mu$ has at least one box. Thus $m < \lambda_i$ and it follows that $\iota(P)$ satisfies the primed row $i$ flag condition. Consequently, $\iota(P)$ satisfies all flag conditions. This shows that $\iota(P) \in PFT(\lambda/\mu)$.

Now suppose that box $b$ of $P$ contains an $m'$. If $\lambda/\mu$ has a box above $b$ then $P$ must have in this box an $s$ or an $s' $ for some $s>m$ by the minimality of $m$ and the minimality of $i$. But it cannot be an $s'$ because column $j$ of $P$ is assumed to be $\prec$. Since $s \prec m$ it follows that column $j$ of $\iota(P)$ is $\prec$.
If $\lambda/\mu$ has a box to the left of $b$ then $P$ must have in this box a $t$ or a $t' $ for some $t \geq m$ by the minimality of $m$. But it cannot be a $t'$ because row $i$ of $P$ is assumed to be $\prec$. Since $t \preceq m$ it follows that row $i$ of $\iota(P)$ is $\prec$. Consequently, all rows and columns of $\iota(P)$ are $\prec$.
Next, suppose that $m>\mu_i$. As stated, if $\lambda/\mu$ has a box to the left of $b$ then $P$ has in this box some $t$ with $t \geq m$. But this would contradict the nonprimed row $i$ flag condition so $b$ must be the leftmost box in row $i$ of $\lambda/\mu$. It follows from this and the fact that row $i$ of $P$ is $\prec$ that the box in row $i$ and column $\lambda_i$ of $P$ contains $n'$ for some $n \geq \lambda_i-\mu_i-1+m$. But this means that $n \geq \lambda_i$ since we assumed $m>\mu_i$. This would contradict the primed row $i$ flag condition so we conclude that $m \leq \mu_i$. This means that $\iota(P)$ satisfies the nonprimed row $i$ flag condition. Consequently, $\iota(P)$ satisfies all flag conditions. This shows that $\iota(P) \in PFT(\lambda/\mu)$.

It is clear that for all $P \in PFT(\lambda/\mu)$ we have $\iota^2(P)=P$ and that $\ell w(P) +rw(P)=\ell w(\iota(P)) +rw(\iota(P))$ and that $(-1)^{\#(P)}=-(-1)^{\#(\iota(P))}$. This proves the lemma.

\end{proof}

\begin{example}
The involution $\iota$ associates the following two tableaux.
\begin{eqnarray*}
\Yboxdim{14pt}
\young(*****544{{3'}}{{5'}},****433{{3'}}{{6'}}{{8'}},****32{{\textcolor{red}{2}}}{{3'}}{{6'}},***32{{2'}}{{3'}}{{4'}})
\,\,\,\,\, \longleftrightarrow \,\,\,\,\,
\young(*****544{{3'}}{{5'}},****433{{3'}}{{6'}}{{8'}},****32{{\textcolor{red}{2'}}}{{3'}}{{6'}},***32{{2'}}{{3'}}{{4'}})
\end{eqnarray*}
In this case, $m=2$, $i=3$ and $j=7$.
\end{example}

\begin{proof}[Proof of Main Fact \ref{mf2}]
Let $\mu \subset \lambda$ be partitions with exactly $\ell$ rows. There is a canonical bijection:
\begin{eqnarray*}
\bigcup_{\mu \subseteq \rho \subseteq \lambda} OFT(\rho/\mu) \times UFT(\lambda/\rho) \rightarrow PFT(\lambda/\mu)
\end{eqnarray*}
It is given by sending a pair $(P,Q)$ to the tableau obtained by superimposing $P$ and $Q$ on a single Young diagram of shape $\lambda/\mu$. Moreover, if $(P,Q) \rightarrow R$ then $\ell w(R)=wt(P)$ and $rw(R)=wt(Q)$ while $\#(R)=|\rho|-|\mu|$ where $\rho$ is the shape of $P$. Therefore writing $T_{\rho}=OFT(\rho/\mu) \times UFT(\lambda/\rho)$, we have:

\begin{eqnarray*}
\sum_{\mu \subseteq \rho \subseteq \lambda} \sum_{(P,Q) \in T_{\rho}}(-1)^{|\rho|-|\mu|}\mathbf{z}^{wt(P)}\mathbf{z}^{wt(Q)}=\sum_{R \in PFT (\lambda/\mu)} (-1)^{\#(R)}\mathbf{z}^{\ell w(R)}\mathbf{z}^{rw(R)}
\end{eqnarray*}
But the latter is $0$ by lemma \ref{z}. This implies the statement of Main Fact \ref{mf2}.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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This week's element is THE PIG, from Traders & Builders (Expansion #2). This otherwise forgotten element from the expansion that brought you trading tiles and Builders gives players a +1 point bonus for one of their Farms. That is, if they can get the Pig on the farm before the game ends! How do you use the Pig? How much success have you had wracking up the points? Describe your dismay when somebody merges the fields, marginalizing or eliminating your bonus. Discuss your relationship with the Pig, as well as your strategies for taking advantage of this expansion.
Tactically I use it to gain an advantage over a shared field. If i'm going to have to share a field with an opponent I'm going to do my best to make sure I can gain a points advantage of some sort over him. I know that this doesn't stop my opponent also putting his pig in the same field but in my experience a lot of people forget that he exists as a piece! More fool them I say!
can be very powerfull if you use it with some other expansions such as Cathars or barns from 5th.
It's been years since I had a normal game. With all the megaCarc we now play, the pig is one of many aspects of a large various collection of elements in the game I tend to forget to use.
But I do make up for it by using the emoticon here quite often.
I quite like the Pig although I've always been confused as to whether it's supposed to lie on it's side (like a farmer) or stand on it's legs like a... well, Pig.
It is certainly easily forgotten but when I remember that it's there I tend to use it in a variety of ways in my games. Sometimes I'll place it on a large, shared farm purely because my opponent(s) has/ have and it feels like we ought to at least get the same number of points from it if we're going to share it. Other times I'll keep it back and place it on a farm of my own. Naturally, it always depends where I can generate the most points.
Using Pigs with Abbey & Mayor is interesting because of the Barn. I remember reading in the CAR that you can place your Pig on a tile that merges an existing farm with a farm with a barn and consequently get 2 points rather than 1 for every city. If this is something you do over and over again it can make a huge difference to your overall score.
All things considered, it goes without saying that this is almost always one of the last pieces of wood to be placed as you don't always know where the most valuable farm is going to be. In this case it's often a race against time to get him down in the right place!
Also, I'd love to see a list of potential ways to annoy your opponent(s) with your Pig. It could provide me with a valuable tactical advantage the next time I play with Traders & Builders!
This is my favorite tactics playing with A&M. I had one game with my wife, where was a huge field (about 15-17 cities), and till the rest of the game we gave up building any more cities, but simply connected to this big field, gaining more and more points. One of us had more luck in connecting and he/she won (don't remember who it was).
If playing also with BC&B, it is even easier mission to complete.
When playing without A&M I simply put a pig to the most valuable farm.
Can a barn and a pig be on the same field? or does it need to be pulled back?
No, it can't, because pig can be on the field only with a follower, but follower can't be on the same field with barn (in case of connect it immediately is scored and taken off the playing field).
This is what I try to do too - it's surely the most obvious and beneficial thing to do with the pig. I'm surprised it took this many posts for someone to say it!
I think that interaction of pig and barn has broken game too much. Really, all other aspect lost value, it was all about joining fields with barns, especially in larger games with multiple expansions.
That's why I hope that future expansions for Carcassone II will be a bit more well tought and designed to integrate with eath other without breaking the game or changing it drastically.
I'll admit that Barn + Pig Field = Lots of Points if done properly. The bigger the field, the bigger the impact.
While I don't plan on buying Carc 2.0, I do admit that the addition of various elements that could be used in future iterations of the game is not shortsighted. I always felt that the little villages in the base game could be used for something, but they are just too small and random. Had the designers thought about it more, they could have made them a bit bigger just in case they wanted to use them in the future. Same with the multi-colored towers in the new game, even if those aren't used now, they could be in the future.
I agree, and I'm looking forward to seeing how these might be utilised. Equally, I'm sure that some of the more creative people here will be able to come up with some of their own fan-expansions that couldn't have been made for standard Carcassonne (old artwork).
I think it's easy to say, with hindsight, that the original game could have included some of the more subtle features like this from the beginning. I guess they just didn't know how successful it was going to be at the time and so wouldn't have seen the point. Who could have predicted that it would sell so well and have 30 or more expansions made for it? I suppose the redesigned artwork gives them the opportunity to plan ahead and ensure that it has plenty of years ahead of it. This can only be a good thing in my opinion, regardless of the problems it causes with regards to completionism (a word?) and backwards compatibility with the old expansions.
Anyway, back to the pig. I remember seeing an idea for a fan-expansion called "The Butcher" which introduced an alternative use for the pig. I can't remember the rules, but they should be hard to find. I'm keen to experiment with some of these once I have the time (and can persuade my wife to play) and would perhaps like to come up with some alternatives of my own.
In case anyone hasn't noticed, I've decided to go though all the Elements of the Week and throw my two cents in where I missed out before. HA!
I tend to forget about the pig. If I have a large farm I'll throw him down, but otherwise he doesn't do much. I tend to play with a fairly light-on farms-strategy, anyway. Mostly because my best girl relies WAY too much on farms. I'll usually trick her into sinking about three followers into a farm just to maintain control of it, then I'll clean up by grabbing every city and road on the table.


In the midst of last Sunday's vegan feast, my friend Tevia (the evening's guest of honor) mentioned that on most mornings she makes vegan pancakes for her little boy (often in the shape of bugs). Tev is an incredible baker so I knew I wanted to get my hands on that recipe. Thankfully, she agreed to share and I've been feasting on these all weekend. So good served with maple syrup and nectarine jam.
Combine the above ingredients (be careful not to overmix). Heat a large skillet or frying pan over medium and grease with coconut or canola oil. Once heated, spoon batter onto the pan and let cook for about two minutes per side, until lightly browned. Enjoy with side of bacon. Just kidding.


The Waterfront Restaurant & Marina, located at the southern tip of Pine Island in St. James City, is a local favorite and sought out by savvy travelers who have heard of its delicious food and casual atmosphere.
The Waterfront Restaurant is a family owned and operated restaurant that serves be freshest seafood and delicious dinners.The Waterfront is the hot spot on the southern end of Pine Island. JD, and the whole Waterfront family invite you to kick back, relax and enjoy a piece of ol' Florida.
Come out early for a table on the deck and watch the sunsets.
Our fresh Florida favorite! Crispy fried chunks with fries or pasta salad.
Pile of jumbo tender clam strips and fries or potato salad.
Bite sized chunks of tender Florida Gator with fries or potato salad.
2 plump, crispy cakes, seasoned with Old Bay and lemon. Served up with our remoulade sauce, corn on the cob and sweet potato fries.
Crunchy Grouper and crisp cabbage fill 2 flour tortillas. Served with a side, Mango Pineapple Salsa, Pico de Gallo and yellow rice.
Tender Baby Back ribs, hand rubber and slow roasted to fall off the bone. Served with corn on the cob and fries on the side.


text:
0 In Fashion/ Interview/ Women Supporting Women
Interview with Catherine "Ketty" Promise : : Co-founder of Artisan Apparel located in Gulu, Uganda
Today I'm so overjoyed to share with you my really good (Insta) friend Ketty. Her and I connected a couple years ago while I was still living on Guam and working on my series Hope For Women in Crisis. I immediately fell in love with the gorgeous dresses she was sharing over on Instagram, and I knew I had to learn more about her brand.
Ketty is one half of the female led label Artisan Apparel, out of Gulu district, Uganda. Her business partner Devon is located here in the U.S. and will hopefully be a guest on the blog soon.
I'd love to hear your thoughts on this interview once you've read it. Please feel free to leave a comment below and visit me over on Instagram.
Catherine "Ketty" Promise, center.
AM: Please introduce yourself.
KP: My name is Okello Catherine Promise (but ever since l could hear, everyone called me Ketty). I was born and raised in a once war torn district called Gulu in the northern part of Uganda in East Africa. I was raised by the most amazing parents, Mr. Okello Santo, ma dad, and Mrs. Ventorina Aciro Okello, ma mum.
I am the first born, with six other siblings, four girls and three boys; but being African, we have lived and grown up with so many cousins. I grew up in an extended family where siblings, aunties, and cousins live together. Ma mother is a housewife and ma dad (RIP) was a builder. They raised us to have Christian values, which I hope I still have and use. My extended family is closely knit, we always have each other's back, and I'm thankful for that.
AM: What is Artisan Apparel Uganda and When did it begin and what made you decide to start it.
KP: Artisan Apparel is a locally owned small business located in Gulu district in Uganda. It was established on May 16th, 2014, but officially launched on March 5th, 2015. It was designed to engage the needs of the community and provide opportunities for advancement in post war Uganda through the unlikely medium of fashion.
The business employs women and (now) two men from the community, while opening up their minds to the creativity that resides within them. We create jobs, design fashion, and empower artisans in Uganda to break the cycle of poverty for themselves, their families, and their communities, by capitalizing on traditional African print fabrics and recycled paper for the jewellery.
AM: What is your role at Artisan Apparel?
KP: I'm the Co-founder/Director. I manage the business side of Artisan Apparel, but ma creative side helps in opening minds and coming up with new fashion ideas that can be incorporated in the business.
AM: Where did you learn to sew and design.
KP: I don't know how to sew but l can design clothing and l didn't have any formal education for this. Its just a passion that l have always had. Our lead tailor Veronica has been sewing for over 20 years and l have always known her. She has always been ma personal tailor.
I would design what I wanted and she would make it a reality– till the day when I told her, we could actually turn this into a partnership and business that would benefit us and the community we are living in.
Ketty and Vero
AM: Did you always know you wanted to be an entrepreneur.
KP: I think in the deep recess of my mind, yes, I always knew I would be an entrepreneur.
I started business at a young age. Ma dad always took me to his building sites and I was always like his apprentice. Ma mum had a small grocery shop that I would attend to during school holidays. Actually ma first start at a business was sponsored by ma dad after I finished university. I opened up a shop that sold second hand shoes and bags… lol see it was still in fashion.
Being Ugandan, we always think after university, one needs a job at an NGO or government entity. So I worked for a bit in several disability organizations, until the contracts ended and being jobless pushed me into pursuing ma passion as a business, while still giving back to the community.
AM: How do you find the artisans that you employ.
KP: The first artisan was Vero, who like I said earlier is more of a partner because we had already been doing this together for awhile. God works in ways best known to Him; when we opened the first artisan shop, it was only me, Vero, Devon, and one other girl that Vero was tutoring at that time. But by the time we officially launched, we had people that wanted to be trained by us and were absorbed into the new Artisan Apparel family, so we started training but with baby steps.
One or two artisans at a time (for the sewing section). For the jewelry, that's another amazing story, because when we started artisan apparel, we didn't want to do paperbead jewelry since so many NGOs were already doing it and already had a market. We thought competing against them would be so hard! But a group of former child solders approached us weeks before the official launch of Artisan Apparel, they already had some skills in making the jewelry, so we made them make samples that we displayed. One of our current partners, Krista Jefferson of iamjustone loved their jewelry and like they say, the rest is history. Lol…so the current artisans bring in more artisans, but we also try to scout for talent within the community so that we can either make them better or promote them.
Its all about building the community.
AM: What is one of the best things about owning your own business?
KP: You can have several worries but one thing is for certain, you can't be worried about someone sacking you the next morning! Lol, but I certainly get absolute joy when paying the artisans for the jobs they do, that to me is best because I know they will not sleep hungry, they will be able to pay school fees for the kids, and be able to just have the basic needs.
AM: What is the most difficult thing about running a business in Uganda?
KP: The attitude that people have towards a new business is very frustrating. You have to work extra hard to prove to them that your product is worth it and that you are capable. It gets worse if you are a woman entrepreneur, but I am glad it is getting better.
AM: Is there any woman who has had a significant impact on your life?
KP: My mother is my mentor, she has instilled in me so many values that have shaped me into what I am now. She is a no nonsense woman, tough, but the kindest person that I know.
Then ma sisters, they are the greatest support base that I have. And of course Devon; she is a angel amidst us all; she has soo much love and compassion, I don't know what I would do without her in ma life. I am such a Catholic so I really do look up to Mother Theresa, bless her soul!
AM: What legacy do you hope to leave?
KP: If we could work together to leave this world a better place, that would be good enough but I wish for people to remember me as a kind and humble soul.
AM: Where do you see Artisan Apparel Uganda in 5 years?
KP: I would love for Artisan Apparel Uganda shop and workshop to be the busiest and most talked about spot in Uganda, and the world. I want everyone that comes into the country to make it a point to shop at Artisan Apparel, but our main goal is to be able to train and employ as many artisans as possible. We want to have a creative space and training centre set up, now that would be the best!
AM: Where can we find you and Artisan Apparel on the web?
KP: I can be found on facebook as Ketty D Promise, on instagram as Ketty Promise and ma email is catherinepromise@yahoo.com.
Artisan Apparel Uganda can be found both on Instagram and Facebook. Our email address is artisanapparelugada@gmail.com, and our website is www.artisanappareluganda.com.
Location in Uganda: Mepaco Building, Plot 9 Market Road after JoJo's Palace Hotel. (Former spot of Cafe Larem).
I hope you LOVED reading this interview as much as I loved doing it! I can't wait to hear your thoughts!
November 1, 2017 By Anjelica
Hey there, I'm Anjelica-- a former Third Culture Kid, turned Global Mama. I grew up traveling the world and now love exploring new places with my husband and our two Little Women. I'm passionate about empowering women to embrace motherhood in their own unique way, sharing the stories of inspiring women from around the world, and offering wisdom on how to live a more conscious and globally-minded life. My family and I recently relocated from Guam to Seattle and we're still adjusting to cooler city life. You can usually find me sipping something warm and caffeinated, chatting about birth and breastfeeding, and looking for other free-spirited babes to connect with. I hope you'll stick around, discover something you love, and say "hey" in the comments. I'm SO glad you're here! Read More...
Anjelica Malone: Providing Culturally-Sensitive Lactation Support
Raw Baltic Amber : : Children's Teething Necklace $22.00
Consultation for Birth Workers $45.00 – $90.00
2017 africa Bali birth book breastfeeding breastmilk business capsule wardrobe career COHI creative crisis designer doctor doula entrepreneur Expectful family global citizen goals Guam guide Japan journal lactation lactation counselor mater mea maternity midwife motherhood podcast postpartum pregnancy Puerto Rico recipe seattle self care texas tips travel Ubud Uganda women yoga

redpajama_set_name: RedPajamaCommonCrawl


An Overlap Tank with in this season's new colorways with an updated, elegant Micro Suede construction.
An easy to toss on tank get a romantic twist with the addition of a ruffled neckline and double spaghetti straps. It features ruffles at neckline and double spaghetti straps.
Loose fit, sleeveless, scoop round neck tank top. Racerback. Has very rounded hems. This tank top is made with medium weight, sand washed knit fabric that has a very "silky" feel, drapes beautifully and stretches very well.
In a lightweight linen-blend this double layered tank featuring an under layer with a racer back and a top scoop tank. It features an unfinished trim, dropped armholes, and is a perfectly distressed.


Most car manufacturers usually design the back windows in a tinted manner, but the front windows in a vehicle remain clear. However, tinted windows have become quite popular these days, and now, both the front and back windows of cars come tinted.
There are numerous benefits of having tinted windows, and you might be familiar with most of them. However, there are some of unknown benefits about tinted windows,which you might know.
One of the main reasons why people choose tinted windows is undoubtedly the privacy it offers. Tinted windows ensure that no one can look into your car from the outside. Furthermore, they also guarantee protection from theft, as crooks will not be able to see the valuables inside your car.
Window tints help to keep the interior of your car to approximately 60 percent cooler. This will prevent you from the distress of entering into a hot car,as well as reduce the risk exposing your skin to the hot air trapped inside the car.
Furthermore, window tints are eco-friendly and you don't have to use much air conditioning either, as window tinting helps to maintain the cool temperature within your car. This means you don't have to use much gas, ultimately saving a lot of money.
The UV rays from the sun that enters into your car through the windows can put frequent drivers at a greater risk. The chances for skin allergies, skin problems, and even skin cancer, are higher when you are exposed to these harmful UV rays.
In addition to that, the harmful UV rays entering your car can also damage the interior of your car. The accessories, dashboard, and cloth upholstery exposed to the UV rays, may gradually fade or start to crack.However, a good quality window tint would keep all these issues at bay by absorbing almost 99 percent of these harmful radiations from the sun.
During an accident, or when an object hits your car window, the window tinting holds the window glass together and prevents it from breaking into small pieces that could get inside the car and harm you. Hence, window tinting keeps you and your family safe from such issues.


Drawing on an experience from my childhood - crisp, white clouds chase along a blue sky, over with white cornices edged, brown, brick walls - I made this video work. The experience of stillness confronted with the passing of the time. See time slipping by: shadows that move across buildings and grasslands. You're irrevocably removed from your origin.
For years I've kept the desire to do something in time-bound media with this experience. In music of J. S. Bach, the in- and outro of a harpsichordconcerto, I recognized the movement of the shadows and clouds. I shot new footage on the flat roof of my former studio-building New Rollecate in Deventer, and combined this - when editing my work in the studio - with old footage shot on the coast of Normandy. I completed the soundtrack with an idea from a yet unrealized short film: the voice of a language course tape which lists time markers.


Mystery shrouds the birth of our universe. In a fraction of a second, the universe transformed from a size smaller than a subatomic proton through expanding exponentially faster than the speed of light, according to the Big Bang theory. At the heart of this event lies the explanation for all the constituents of the cosmos today. If the moment of the Big Bang can be understood, we may finally have the theory of everything that can reconcile the quantum Standard Model of particle physics with Einstein's general theory of relativity, which holds that gravity is a result of the curvature of space and time.
Even the most powerful particle physics experiments on Earth don't have enough clout to recreate the conditions in the early universe. We can find evidence of them, however, in the cosmic microwave background (CMB), which functions like a fossil remnant of that very early universe, says Rachel E. Bean, Astronomy.
In an effort to understand physics at those extreme properties, Bean looks for tiny temperature fluctuations in the CMB. These were generated a trillionth of a second after the Big Bang during a process called primordial inflation when the universe is thought to have expanded faster than the speed of light for a brief time. "We have to describe how quantum properties behave with gravity and space and time," says Bean. "We don't know how to do that. The only way we can try to figure this out is to look at these very early moments." Bean hopes to connect these temperature fluctuations in the CMB to one of the potential theories—especially string theory—that are candidates to reconcile quantum mechanics and the general theory of relativity.
The CMB can also tell scientists about the effects of gravity on objects through time.
Astronomers will be able to do that in unparalleled detail over the next decade when four new telescopic surveys come online. These large-scale structure surveys will look at millions to billions of galaxies. They will either take multicolor images of them, revealing through color different physical properties, or they will record the galaxies' spectra, the emission or absorption lines of light of particular wavelengths, which pinpoint the galaxies' positions in space. "We're going to be able to survey out billions of light years to be able to understand the structure of our universe with unprecedented precision," Bean says.
Two of the telescopes will be ground-based and two will be space-based. Bean is the leader of an international collaboration of approximately 500 scientists—the LSST Dark Energy Science Collaboration—that will be using the data from one of the ground-based photometric imaging telescopes, the Large Synoptic Survey Telescope (LSST), which is an international venture led by two United States agencies, the National Science Foundation and the United States Department of Energy.
The LSST will share the ground-based spotlight with another state-of-the-art telescope, the Dark Energy Spectroscopic Instrument (DESI), a United States Department of Energy initiative, which will come online in 2019.
The final two telescopes, both space-based, will be launched in the next decade: the European Space Agency's Euclid in 2021 and the National Aeronautics and Space Administration's Wide Field Infrared Survey Telescope (WFIRST) in the mid-twenties.
Euclid, ESA's Dark Universe mission, has passed its preliminary design review, providing confidence that the spacecraft and its payload can be built. It's time to start 'cutting metal'.
First proposed to ESA in 2007, Euclid was selected as the second medium-class mission in the Cosmic Vision programme in October 2011. Italy's Thales Alenia Space was chosen as the prime contractor in 2013.
Since then, the mission's design has been studied and refined. This has involved a wide range of detailed technical designs, in addition to building and testing key components.
The outcome of Euclid's recent review was positive, opening the door to the industrial contractors and external instrument teams building the spacecraft and payload for real. Airbus Defence & Space in France will deliver the complete payload module incorporating a 1.2 m-diameter telescope feeding the two science instruments being developed by the Euclid Consortium.
"This is a major milestone for us. Everyone is now ready to start cutting metal," says René Laureijs, Euclid's project scientist.
On the scientific side, this review checked that the mission can indeed deliver the required data. The combined performance of the spacecraft, telescope and instruments shows that the data returned over the six-year mission will achieve the objectives.
Euclid is designed to give us important new insights into the 'dark side' of the Universe, namely dark matter and dark energy, both key components of the current model for the formation and evolution of the Universe.
Observations made over recent decades reveal that less than 5% of the matter in the Universe is in the form of normal atoms, while a much larger amount of dark matter is inferred from measurements including the rotation speeds of galaxies. This matter acts through gravity, but is invisible.
Dark energy, on the other hand, is invoked to explain the finding that the expansion of the Universe is accelerating.
Although they are thought to make up the majority of the matter and energy in the Universe, dark matter and dark energy cannot be seen. Instead, their presence is inferred by the movement of galaxies, the shape of galaxies, their distribution in space, and the rate of the Universe's expansion as traced by the galaxies.
By mapping the shapes, positions and movements of two billion galaxies across more than a third of the sky, Euclid will provide astronomers with an unprecedented wealth of data to analyse.
The unrivalled accuracy of its measurements will allow them to close in on the properties and behaviour of dark matter and dark energy. This, in turn, will put constraints on the theoretical properties of what these two unseen components of the Universe may be.
With Euclid's preliminary design review now safely passed, the next major milestone comes in two years at the critical design review.
At this point, the major hardware components will have been built and tested. If all goes well, Euclid will then be assembled.
After this, Euclid will be ready for launch in December 2020 on a Soyuz rocket from Europe's Spaceport in Kourou, French Guiana.
"The construction of ESA's Euclid space mission to explore the 'dark Universe' will be led by Italy's Thales Alenia Space as prime contractor, beginning the full industrial phase of the project.
The announcement follows that of last month when Astrium Toulouse was confirmed to build the payload module – the telescope and optical bench carrying the science instruments.
Euclid will be launched in 2020 to explore the roles played by dark energy and dark matter in the evolution of the Universe since the Big Bang and, in particular, in its present accelerating expansion.
Dark matter is invisible to normal telescopes, but acts through gravity to play a vital part in forming galaxies and slowing the expansion of the Universe.
Dark energy, on the other hand, causes a force that overcomes gravity and that is accelerating the expansion seen around us today.
Together, they are thought to comprise 95% of the total amount of mass and energy in the Universe, with 'normal' matter – from which stars, planets and we humans are made – making up the remaining small fraction. But their nature remains a profound mystery.
'We are pleased to confirm the prime contractor for this exciting mission. With the support of European space industry, we are a step closer to revealing the darkest secrets of the Universe,' says Professor Alvaro Giménez, ESA's Director of Science and Robotic Exploration.
'This is a long-awaited milestone after the mission concept was first proposed to ESA in 2007, and we are delighted to see that the spacecraft construction can now begin,' says Yannick Mellier, who leads the Euclid consortium, comprising scientists from 13 European countries and the US.
The consortium will provide Euclid's two state-of-the art scientific instruments: a visible-light camera and a near-infrared camera/spectrometer. Together, they will map the 3D distribution of up to two billion galaxies spread over more than a third of the whole sky.
Light from the most distant galaxies streaming towards Earth is slightly bent by gravity as it interacts with matter along the way. This is dominated by dark matter, whether associated with galaxies and galaxy clusters, or in isolation. Thus, by measuring distortions in the shapes of those background galaxies, astronomers can construct a 3D map of the dark matter in the Universe.
Furthermore, by assessing how the distribution of galaxies and galaxy clusters has changed over cosmic time, scientists can infer the role and evolution of dark energy from the dawn of the Universe until today.
See the full article http://www.esa.int/Our_Activities/Space_Science/Thales_Alenia_Space_kicks_off_Euclid_construction.
"The European Space Agency (ESA) has selected three NASA-nominated science teams to participate in their planned Euclid mission, including one team led by NASA's Jet Propulsion Laboratory in Pasadena, Calif.
NASA is a partner in the Euclid mission, a space telescope designed to probe the mysteries of dark energy and dark matter. Euclid is currently scheduled to launch in 2020.
JPL will provide 16 advanced infrared detectors and four spare detectors for one of two instruments planned for the mission. In addition, JPL will contribute to science planning and data analysis with the help of its 43-member science team, the largest of the three U.S. teams. This team, led by JPL scientist Jason Rhodes, is composed of 29 scientists recently nominated by NASA, and 14 U.S. scientists who are already part of Euclid.
Rhodes also was appointed by NASA to be a member of ESA's principal 12-member Euclid Science Team and the U.S. representative for the Euclid Consortium's governing body. The Euclid Consortium is an international body of 1,000 members, including the U.S. science team members, and will build the instruments and analyze the science data jointly.
Euclid will observe up to two billion galaxies occupying more than one-third of the sky with the goal of better understanding the contents of our universe. Everyday matter that we see around us, for example in tables and chairs, people and even stars, makes up only a few percent of everything in our cosmos. If you could fill a bucket with the mass and energy contents of our universe, this everyday matter would fill only a small fraction. A larger amount, about 24 percent, would consist of dark matter, an invisible substance that does not reflect or emit any light, but exerts a gravitational tug on other matter.
The other two U.S. science teams are led by Ranga-Ram Chary of the Infrared Processing and Analysis Center at the California Institute of Technology, Pasadena; and Alexander Kashlinsky of NASA's Goddard Space Flight Center, Greenbelt, Md.; with three and seven members, respectively.
"NASA has officially joined ESA's Euclid mission, a space telescope designed to investigate the mysterious natures of dark matter and dark energy.
To be launched in 2020, Euclid's 1.2 m-diameter telescope and two scientific instruments will map the shape, brightness and 3D distribution of two billion galaxies covering more than a third of the whole sky and looking back over three-quarters of the history of the Universe.
Scientists hope to solve key problems in our understanding of the evolution and fate of our expanding cosmos: the roles played by 'dark matter' and 'dark energy'.
NASA recently signed a Memorandum of Understanding with ESA outlining its contribution to the mission. The US agency will provide 20 detectors for the near-infrared instrument, which will operate alongside a visible-wavelength camera. The instruments, telescope and spacecraft will be built and operated in Europe.
NASA has also nominated 40 US scientists to become members of the Euclid Consortium, who will build the instruments and analyse the science data returned from the mission. The consortium already includes almost 1000 scientists from 13 European countries and the US.
Dark matter, unlike most of our own planet's little "mysteries," is a real enigma with big potential.


Owners of Highlands NC real estate who are considering selling, how wired is your home? Today's technology driven world makes owning a second home more and more appealing as people can get work done wherever they are and enjoy a better lifestyle at the same time. Here in Highlands NC many people work during part, if not all, of their time here and the more tech-friendly your home is the more appealing it will be to potential buyers.
If your home is not up-to-date with technology it can be. Consider setting an appointment with a specialist who can make your home more efficient for the internet as well as making wireless activity a breeze. If you can tout your home's high tech abilities among its accolades you could be more attractive to would-be buyers.
Contact me, Lynn Kimball, today for more information of selling Highlands real estate and how you can get your home ahead of the competition. Real estate is picking up speed and marketing the attributes of your home can help your home attract more potential buyers through your front door.


The Physical Plant is charged with providing clean, safe, and properly equipped facilities on the Boston Conservatory at Berklee campus. For more information regarding Physical Plant, including maintenance and custodial services, sustainability, and staff, please log in to Berklee's internal website. For general maintenance, custodial, and event setups please submit a work order form.
Key requests and lock changes are processed by Public Safety. Please contact Lance Finamore at 617-747-2587 or lfinamore@berklee.edu.
If you are looking to rent a Conservatory facility, contact the Production Office.
For after-hours emergencies, please contact Public Safety at 617-747-2682.
Physical Plant Office: Monday–Friday, 8:00 a.m. to 5:00 p.m.
Custodial: Monday–Friday, 6:00 a.m. to 11:30 p.m.
Maintenance: Monday–Friday, 7:00 a.m. to 8:00 p.m.


The scope of works for this job in Berwick 3806 included tidy up cabling, wall mount 50" plasma installation, install network cable for the use of Smart TV and reinstate a previous second zone audio room.
The mess of cables were all disconnected and significantly reduced. New wall plates installed. Complicated user unfriendly set up of equipment was reduced to only 2 HDMI inputs. No audio working to the second zone billiard room was being caused by incorrect wiring – mono ceiling speakers connected to the stereo 8ohm inputs of the amplifier causing the amp to overload and turn off.
The room is now more open and stylish. Neat cabling allows ease of access for cleaning. The entertainment system is now more user friendly with access to Facebook, YouTube and other Smart TV applications. The sound system very clear and impressive with added volume control in the second zone.
Moved House – No Television Reception?
To request a quick quote, please fill-out the below form with as much detail as possible.
Hayes TV have been in the antenna, cabling and home entertainment business for over 20 years. We have installed hundreds of digital antennas around South East Melbourne. We also install NBN network cabling and setup home theatre systems and other home entertainment projects. We are available 7 days a week.
We service all of Melbourne's South Eastern Suburbs. For example, City of Casey, Berwick, Narre Warren, Cannons Creek, Clyde, Endeavour Hills, Hallam, Nar Nar Goon, Hampton Park, Harkaway, Tynong, Lang Lang, Cranbourne, Cardinia, Beaconsfield, Koo Wee Rup, Emerald, Lysterfield South, Officer, Pakenham and more.


Tuesday to Saturday 9am to 6pm.
We strive to deliver the highest quality product to our customers, we source from the finest farms and crafts people. If you are not 100% happy with your order we will make things right with a full refund.


text:
On "Ted"
From The Dispatch:
It's a bold, raucous comedy that takes no prisoners and shows no mercy, emerging as something both hilarious and yet strangely touching, sustaining its rollicking energy for its entire running time. Not only is it a stroke of comedic genius, "Ted" is one of the funniest movies to come along in years.
Click here to read my full review.
"Aurora" By Hans Zimmer
Review | "Where Do We Go Now?"
Blu-ray Review | "The Turin Horse"
On "The Dark Knight Rises"
Blu-ray Review | "Gray's Anatomy"/"And Everything ...
Blu-ray Review | "The Saphead"
On "The Amazing Spider-Man"
Blu-ray Review | "Once Upon a Time in Anatolia"
Blu-ray Review | "The Devil's Needle & Other Tales...
Blu-ray Review | "Little Lord Fauntleroy"


text:
Q: javascript scrollbars without mousewheel support I have some child divs that may or may not be scrollable. I would like for scrollbars to appear if it is scrollable. However I would like these bars to be usable only with click and drag and for the mousewheel to scroll the parent div. I've been looking at some jquery scrollbar plugins but they all support mousewheel and will allow control of the parent only when the scroll limit of the child has been reached. Does anyone know of a jquery (or other) plugin that will support this behaviour?
Any help greatly appreciated
C

A: I believe that adding a mousewheel event, and preventing the default behaviour should work, i.
$('#child').on('mousewheel', function (e) {
    e.preventDefault();
});

If you use:
overflow:auto;

the scrollbars will only appear when needed.
See this jsFiddle example.
I hope that this helps.

redpajama_set_name: RedPajamaStackExchange


2/5/2019: Our internet and phone services are up and running! Please call (212) 265-7250 to reach out office and schedule appointments, consultations, or discuss your dental health with Dr. Sofer.
Contact Dr. Robert Sofer for your New York dentist, New York family dentist, and New York cosmetic dentist needs.


The famously friendly city of Glasgow is stepped in history as one of the great Victorian centres of trade and manufacturing. Scotland's biggest city, Glasgow is a super-stylish mishmash of old and new.
'The Second City of the Empire' is now one of the UK's hippest spots, with plenty of chic bars, trendy cafes and bouncing music spots.
Get the 'local low-down' on Glasgow with insight into great things to see and do, as well as recommended places to eat, drink and dance in the 'Dear Green Place'.
A seamless week's feed of content, providing daily choices of restaurants, attractions and events in Glasgow, as well as a comprehensive spotlight on the best upcoming high-profile concerts, productions and more.
Discover and book tables, takeaways, tickets, hotels, taxis and more in Glasgow, all in the same place without leaving the app, or create and share plans to help maximise your time with friends.


text:
mvb returns
mvb returned to further explore her (hungry) submission to My (bountiful) Domination.
Because mvb is so committed to the experience, it was effortless for Me to respond in kind.
I slipped into Domspace very quickly. And there I continued to flow for three hours.
…carefully dressing her to My satisfaction…
….an exhilarating spanking which left Me out of breath — and mvb crying out "YES! YES! YES!"…
…delicious clit and ball torture…
…to her need to be dragged down, beaten, and fucked into…freedom.
…and the next day, she had this to say:
Oh wow.
i'm 110% certain that i've never had a more exhilarating, intense, glorious, and wholly fulfilling experience in my entire life. You *get* me, and because of that, i felt this feeling of complete emotional and physical submission Yesterday that i've never felt before. i wasn't necessarily scared by it. Instead, it felt like this cathartic, evolutionary thing, this almost "caterpillar-to-butterfly" feeling.
There's a thing that happens when You touch me in any way that i can't explain as feeling anything other than perfect. From here on out, if You're giving, i'm accepting, lol. i'm here for all of the emotion, all of the growth, all of the little tiny explosions of endorphins that make me feel like i'm blasting deeper and deeper into my best self while in Your hands. There's a real honesty and truth in what You do that's really powerful in the way that i feel if i give You more of myself, then You're going to somehow make that more of me just shine and sparkle.
i've spent the last 18 hours laughing out loud for minutes on end and feeling my body just spasm out of the blue in the best ways possible. i also blush at random times when i think about the idea that You literally fucked me silly. In doing that, You made a dream of mine come true.
Here's some full disclosure that i feel really comfortable now about letting You know (though You probably already know this), lol. i'm a slut. Like, in my mind, the sluttiest of sluts. The most depraved of sluts. But, i also view this as this really empowered thing for me where i like, create safe space for myself. There's a thing about everything involved in my own identity and submission (and how they relate and interact with each other) that for me feels like freedom and discovering a peace of mind that makes me feel whole. It's like, the biggest and heaviest thing in my life. You make it feel lighter than air.
Honestly, i spend more time than i'm comfortable telling most anyone reading about and watching documentaries about my favorite femme fatales, then doubling down on watching really extreme transsexual and Woman-on-woman Femdom porn. Like, almost obsessively sometimes. Then, i try to find the happy place between all of these images, actions, and myself. That's my greatest frustration these days and so much of why O/our sessions are so amazing to me. i present my body to You, and you present back to me this whole other person that You've like, used and abused in the best way and then put a bow on top like, "here's who You *actually* are." It's so great, so liberating, and so beneficial. Like, a giant weight has been lifted off my shoulders and i just feel so unencumbered in all facets of my life because of that. Thank You for all of this.
Author Mistress TissaPosted on February 20, 2017 Categories UncategorizedTags bondage, CBT, corporal punishment, Domspace, femme play, gender, Mistress Tissa, mvb, RBP, reviews, rope bondage, rough body play, sessions, spanking, transformative BDSM
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Viratec has a highly experienced team around PHP technologies. We provide development services around various PHP frameworks for our clients in Nigeria and our service including Drupal, CakePHP, Zend, Magento etc. With an experienced team of PHP developers, Viratec has effectively completed numerous PHP projects including mobile applications, CMS driven websites, CRM solution, community tools, discussion forums and many more.


Since I have to have this thing removed and I have suffered with endometriosis and ME, how do I arrange to have the big 'U' shipped for research. Do I just email WPI or Klimas or who? It would be a shame to have a perfectly good organ go to waste. If xmrv is implicated in the prostrate, could it not do something to the uterus? Your replies are most welcome.
Wendy, contact the WPI's biobank and tell them you are interested in donating tissue. They can probably give you and idea of if they need it, how to have your surgeon/pathologist perserve it, and how to ship it.
I checked the link and will email them. Thanks a bunch!
Wendy I thought you were offering surrogacy services for CFS couples!!! I hope your uterus will find a good home for scientific purposes. I also hope that the other organs in your body won't be too lonely with out it. Hoping for a safe surgery and recovery.
This is my favorite thread title ever.
I agree with Anne, such a great thread title. I did not know what to expect, but I did not expect the thread to be serious.
However, after reading I thought.....wonderful idea... wonderful attitude to be considering how to help this cause.
Has your surgery been scheduled?
The other positive thing is the psychiatrists can no longer say you are "hysterical" after the hysterectomy.
Thank you for donating your organ.
Please have a safe procedure.
If you would, when WPI responds to your kind offer, could you also tell us too please? I too will be in the position to offer up my miserable uterus in the next two months. I had visions of throwing it against the wall 360 times to make up for all the pain and problems it has created for me, but, if WPI will actually take the darn thing, than I would give mine to them as well. I sort of doubt they will take body organs such as this, but, hey, you never know. Good idea.
Interesting topic, Wendy, & I hope the "U" goes to a good home, where it can feel useful.
I had an endometrial ablation late December. Destroys the whole lining of the (abnormal) uterus & the pathology taken on the tissue removed before the ablation, turned out to show no cancer, so I was overwhelmingly relieved. My symptoms were the type normally indicitive of endometrial cancer.
Endometrial ablations take 90 seconds with a special laser & this relatively new procedure was done instead of a hysterectomy.
So far, so good. Symptoms are gone 100% & I have had no after effects which commonly occur in the first month or two.
Were you given the option of a endometrial ablation at all? Or should I say, WOULD an endometrial ablation have solved your problems?
No need to give details, I just wondered whether my procedure was an option for you.
There are times when I want to keep my parts because I think it's meant to be there - not that I've ever used it for it's true purpose. And there are times (many many times) that I curse and loathe it's existence, would rip it out with homemade utensils if I could and run it over will my car!!!!
I say this every month: My uterus is not my friend!
A friend of mine had surgery a few months ago (not a hysterectomy) and contacted WPI. They did want samples, she has a fairly unique set of issues.
Anyway, the important thing to share is that she thought she had everything in place, and then on day of surgery found that her hospital was almost hopelessly unprepared to make it happen. In addition to dealing with the stress of surgery day, she practically had to do the hospital's job to make sure that her bone and tissue specimens got correctly collected, packaged, and shipped.
Thank you for sharing your girly parts for the greater good, and I wish you a safe and successful procedure and speedy healing. I wish there would've been someplace to send samples of my practically useless parts when I was relieved of them.
Maybe you want to contact the CAA also, since they're getting a biobank going? I would think there will be plenty to go around, my understanding is only samples get sent, not the whole enchilada (uter-lada?).
" she practically had to do the hospital's job to make sure that her bone and tissue specimens got correctly collected, packaged, and shipped."
SeaShel, and others, I am postop bunionectomy, sitting in bed with my foot on a tower of pillows. Would my bunion bone fragments have been of interest?????
I hope you heal well and swiftly - take good care.
This thread has made my day for all sorts of reasons.
What a brilliant community we have.
This is such a brilliant idea! I certainly hope it happens because it would take nothing but willpower and a little organisation and I would think that you Wendy would be doing the hardest part! Well, bar the surgeons anyway! This is an absolutely wonderful thing you want to do and a brilliant thread title.
On a related note, I wonder how useful pap smears would be to the WPI? In England you get a 'Happy 25th Brithday from thr NHS!' with the offer of a cervical smear test, huzzah how kind! There's no one really prepared here and with XMRV at such an early stage I doubt the Ramsay tissue bank they are forming would be interested just yet because it ain't cheap. I wonder whether the WPI would be though?
Hope your op goes smoothly & your recovery is swift.
My Gosh, so many replies, I do feel my sense of humor returning, although I can't follow alot of the science, brain fog has taken me over. I wonder if Rip Van Winkle suffered from ME/CFS?
Kati, thank you but no surrogate here, I am sure that I and my other organs won't be missing it!
Anne, thank you, I did think hard about a funny post, esp after 'Are we potatos'?
IntuneJune, again, as always, a sincere thank you and I am trying to schedule surgery around school.
Hope 123, thank you for your link, I will use it with my doctors.
Alesh, I agree, no longer hysterical but still, I hope, funny, if not hysterically so.
Sam, thank you for your very kind words and wishes.
Muffin, great name by the way, in NH we are very fond of such nick-names. I too am glad to be rid of the damn thing. I am hoping it goes to a good cause and not to Hannibal lecter!
Thank you all, what a great place!
So many more replies in the time it took me to thank all of you! What a wonderful bunch of people here and, if I can be a bit sexist, a great group of supportive women! I am truly blessed. I will take all of you advice to heart and will do my utmost to contribute to the research of us all. 'God bless us everyone!"


text:
Barack Obama and Joe Biden Join Forces in Video Targeting Trump | The New York Times
Robert E. Lee High in Virginia Will Be Renamed for John Lewis, District Says | The New York Times
The former president and vice president released a new video Thursday meant to cast President Trump as unbefitting of the office, while trumpeting the credentials of Mr. Biden.
By EJI Staff, The New York Times
Former Vice President Joseph R. Biden Jr. joined forces with his top surrogate Thursday morning, releasing a video of a conversation with former President Barack Obama that cast the current occupant of the White House as unworthy and Mr. Biden as the perfect leader to replace him.
Mr. Biden and Mr. Obama covered several topics in the wide-ranging, 15-minute conversation, including President Trump's faltering response to the coronavirus pandemic, health care, the economic recovery, police brutality and presidential leadership.
Racism Informs the Lies of American Exceptionalism | ZORA
The conversation, recorded at Mr. Obama's Washington office, was "socially distanced," as Mr. Biden continues to contrast himself with Mr. Trump, who has only halfheartedly embraced virus mitigation tactics like wearing face masks and staying six feet away from another person. Mr. Biden and his former boss entered the office wearing masks before sitting in chairs across the room from each other, unmasked, as they discussed how Mr. Biden would govern as president.
Featured Image, Joe Biden Campaign Ad, YouTube
KOLUMN Magazine July 24, 2020


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text:
use proc_macro2::Span;
use syn::{ext::IdentExt as _, spanned::Spanned as _};

use crate::common::parse::TypeExt as _;

/// Parses downcasting output type from the downcaster method return type.
///
/// # Errors
///
/// If return type is invalid (not `Option<&OutputType>`), then returns the [`Span`] to display the
/// corresponding error at.
pub(crate) fn output_type(ret_ty: &syn::ReturnType) -> Result<syn::Type, Span> {
    let ret_ty = match &ret_ty {
        syn::ReturnType::Type(_, ty) => &*ty,
        _ => return Err(ret_ty.span()),
    };

    let path = match ret_ty.unparenthesized() {
        syn::Type::Path(syn::TypePath { qself: None, path }) => path,
        _ => return Err(ret_ty.span()),
    };

    let (ident, args) = match path.segments.last() {
        Some(syn::PathSegment {
            ident,
            arguments: syn::PathArguments::AngleBracketed(generic),
        }) => (ident, &generic.args),
        _ => return Err(ret_ty.span()),
    };

    if ident.unraw() != "Option" {
        return Err(ret_ty.span());
    }
    if args.len() != 1 {
        return Err(ret_ty.span());
    }

    let out_ty = match args.first() {
        Some(syn::GenericArgument::Type(inner_ty)) => match inner_ty.unparenthesized() {
            syn::Type::Reference(inner_ty) => {
                if inner_ty.mutability.is_some() {
                    return Err(inner_ty.span());
                }
                inner_ty.elem.unparenthesized().clone()
            }
            _ => return Err(ret_ty.span()),
        },
        _ => return Err(ret_ty.span()),
    };
    Ok(out_ty)
}

/// Parses context type used for downcasting from the downcaster method signature.
///
/// Returns [`None`] if downcaster method doesn't accept context.
///
/// # Errors
///
/// If input arguments are invalid, then returns the [`Span`] to display the corresponding error at.
pub(crate) fn context_ty(sig: &syn::Signature) -> Result<Option<syn::Type>, Span> {
    match sig.receiver() {
        Some(syn::FnArg::Receiver(rcv)) => {
            if rcv.reference.is_none() || rcv.mutability.is_some() {
                return Err(rcv.span());
            }
        }
        _ => return Err(sig.span()),
    }

    if sig.inputs.len() > 2 {
        return Err(sig.inputs.span());
    }

    let second_arg_ty = match sig.inputs.iter().nth(1) {
        Some(syn::FnArg::Typed(arg)) => &*arg.ty,
        None => return Ok(None),
        _ => return Err(sig.inputs.span()),
    };
    match second_arg_ty.unparenthesized() {
        syn::Type::Reference(ref_ty) => {
            if ref_ty.mutability.is_some() {
                return Err(ref_ty.span());
            }
            Ok(Some(ref_ty.elem.unparenthesized().clone()))
        }
        ty => Err(ty.span()),
    }
}

redpajama_set_name: RedPajamaGithub


There are numerous reasons for a person to eat out and visit a restaurant, and the proof that we do so is proven by the multi-billion dollar restaurant chain industry. For some of us, going out is a celebration; perhaps a birthday, or a promotion at work. For others, it could be a way to share a romantic evening or to catch up with friends.
Restaurants are wonderful in that they provide us with good-tasting food, in spite of our predispositions in the kitchen, and does so quickly, conveniently, and hassle free. While some of us are certainly more apt in the kitchen then others, restaurants allow everyone to eat their fill and go on with their lives. However, whether you are going to a restaurant out of convenience, to celebrate, or simply to indulge, it is important to know that not all restaurant chains are created equal.
The word 'chain,' when associated with a restaurant, typically doesn't bring the best notions to mind, at least in terms of quality of food. Knowing as we do, these are businesses, and ultimately, they are after making a profit, a profit which can be much better severed by buying mass produced foods, and in many cases, simply reheating and 're-assembling' them in house. But just as, there are many poor restaurants that using the word food to describe their offerings might be a stretch, there is a good amount of others that seem to offer a healthy alternative – at least by comparison.
While it cannot be argued that if you are trying to eat healthy than eating at home is your smartest move, however, sometimes eating out is the only option on the menu. So next time you do find yourself in a position to pick the restaurant, may we suggest one of these 10, because compared to the rest, these are the 10 healthiest restaurant chains.
This should come as no surprise, but the answer is yes, this is a chain of restaurants based around the all mighty noodle. While they do offer a little bit of everything, like soups, salads, sandwiches and desserts, noodles are their bread and butter, so to speak.
From spaghetti and meatballs, to pad Thai and everything between, this menu has something for everyone and even allows you to customize your meals. Unlike the power of doing that with a pizza, at Noodles & Company, it provides the patron with a lot of health menu options, so that they, in fact, can feel good after eating at a place called Noodle & Company.
Fact: Only soybean oil is used for sautéing at this chain, and lean proteins include hormone- and antibiotic-free chicken, beef, and shrimp and organic tofu. Whole-grain linguine makes for a healthy noodle option, and many small soups and salads contain 250 calories or fewer per serving.


The traditional bowl of rice is, by far, the most important food in the world today. Half the world's population eats it three times a day.
Rice is one of the most nutritious of grains, low in fat and sodium and high in fiber. It is composed of about 80% carbohydrates and contains small amounts of protein, vitamin B1, phosphorus, and potassium. Brown rice is more nutritious than white, which is primarily starch. It retains its oil-rich germ and bran and is packed with vitamins. It also contains fiber and protein.
Basmiti rice is a long-grain, aromatic variety. To develop its full flavor, this rice is aged for at least a year after harvest. It doubles its length while cooking.
All types of rice are high-carbohydrate, low-fat, low-sodium foods that supply iron and many of the B vitamins.
Use rice in breads, cereals, side dishes, and soups.
The secret for good rice is to use a pressure cooker. Brown rice cooks in about 20 minutes and is amazingly fluffy.