text:
\section{Introduction}
\noindent
A {\em labelled poset} $(P,\omega)$ is an $n$-element poset $P$ with
a bijection $\omega\ :\ P\rightarrow[n]=\{1,2,\ldots,n\}$
called the {\em labelling} of $P$. A {\em linear extension}
of $P$ is an order-preserving bijection $f\ :\ P\rightarrow[n]$.
It is natural to define the {\em sign} of $f$ as
$-1$ to the power of the number of inversions with respect to
the labelling, i.e.,~pairs $x,y\in P$ such that
$\omega(x)<\omega(y)$ and $f(x)>f(y)$.
The {\em sign-imbalance} $I_{P,\omega}$ of $(P,\omega)$ is the sum of the
signs of all linear extensions of $P$.
Note that $I_{P,\omega}$ is independent of the labelling $\omega$
up to sign. In this paper we will mainly discuss the square
of sign-imbalances, and then we may drop the $\omega$ and write
$I_P^2=I_{P,\omega}^2$.

If $I_P^2=0$ the poset is {\em sign-balanced}. Such posets have
been studied since 1989 by F.~Ruskey \cite{ruskey2}, \cite{ruskey1},
R.~Stanley \cite{stanley}, and
D.~White \cite{white}.
It is a vast subject however, and most of the work has been
devoted to a certain class of posets: the partition
shapes (or Young diagrams). Though no one so far has been able to
completely characterize the sign-balanced partition shapes,
this research direction has offered a lot of interesting results.
Many people have studied the more general notion of
sign-imbalance of partition shapes, among those
T.~Lam \cite{lam}, A.~Reifegerste \cite{astrid},
J.~Sj\"ostrand \cite{jonas},
M.~Shimozono and D.~White \cite{shimozonowhite},
R.~Stanley \cite{stanley}, and
D.~White \cite{white}.

Young tableaux play a central role in the
theory of symmetric functions (see~\cite{fulton}) and there
are lots of useful tools for working with them that are
not applicable to general posets. One outstanding tool is the
Robinson-Schensted correspondence which has produced nice results
also in the field of sign-imbalance, see \cite{jonas}, \cite{astrid},
and \cite{shimozonowhite}.

As suggested in \cite{jonas} a natural step from partition shapes
towards more general posets would be to study {\em skew}
partition shapes. They have the advantage of being surrounded by
a well-known algebraic and combinatorial machinery
just like the ordinary shapes, and possibly they might shed some light
on the sign-imbalance of the latter ones as well.
We will use
a generalization of the Robinson-Schensted algorithm for skew
tableaux by B.~Sagan and R.~Stanley \cite{saganstanley}.

In a recent paper~\cite[Theorem~4.3 and~5.7]{frank}
E.~Soprunova and F.~Sottile
show that $|I_{P,\omega}|$ is a lower bound for the number
of real solutions to certain polynomial systems. Theorem~6.4
in~\cite{frank} says that $|I_{P,\omega}|$ is
the characteristic of the Wronski projection on certain projective
varieties associated with $P$. When $P$ is a skew partition shape
this is applicable to skew Schubert varieties in Grassmanians
(Richardson varieties).

An outline of this paper:
\begin{itemize}
\item After some basic definitions in section~\ref{sec:prel},
  in section~\ref{sec:skewrs}
  we briefly recall Sagan and Stanley's skew RS-correspondence
  from~\cite{saganstanley}.
\item
  In section~\ref{sec:results}
  we state our main results without proofs
  and examine their connection to old results.
\item
  In section~\ref{sec:mainproof} and ~\ref{sec:inoutproof}
  we prove our main theorems through
  a straightforward but technical analysis.
\item
  In section~\ref{sec:specialisations} we
  examine a couple of interesting corollaries
  to our main results. One corollary is
  a surprising formula for
  the square of the sign-imbalance of any ordinary shape.
\item
  Finally, in section~\ref{sec:future} we
  suggest some future research directions.
\end{itemize}

\section{Preliminaries\label{sec:prel}}
\noindent
An (ordinary) {\em $n$-shape} $\lambda=(\lambda_1,\lambda_2,\ldots)$ is
a graphical representation (a Ferrers diagram)
of an integer partition of $n=\sum_i\lambda_i$. We write
$\lambda{\,\vdash\,} n$ or $|\lambda|=n$. The {\em coordinates} of a cell
is the pair $(r,c)$ where $r$ and $c$ are the row and column indices.
Example:
\begin{center}
  \setlength{\unitlength}{0.5mm}
  \begin{picture}(50,40)(0,-45)
    \put(-70,-27){$(6,4,2,2,1)\ \ =$}
    \put(0,0){\line(1,0){60}}
    \put(0,-10){\line(1,0){60}}
    \put(0,-20){\line(1,0){40}}
    \put(0,-30){\line(1,0){20}}
    \put(0,-40){\line(1,0){20}}
    \put(0,-50){\line(1,0){10}}
    \put(0,0){\line(0,-1){50}}
    \put(10,0){\line(0,-1){50}}
    \put(20,0){\line(0,-1){40}}
    \put(30,0){\line(0,-1){20}}
    \put(40,0){\line(0,-1){20}}
    \put(50,0){\line(0,-1){10}}
    \put(60,0){\line(0,-1){10}}
    \put(38,-32){$(3,2)$}
    \put(35,-30){\vector(-4,1){20}}
  \end{picture}
\end{center}

A shape $\mu$ is a {\em subshape} of a shape $\lambda$ if
$\mu_i\leq\lambda_i$ for all $i$.
For any subshape $\mu\subseteq\lambda$ the
{\em skew} shape $\lambda/\mu$ is $\lambda$ with $\mu$ deleted.
A {\em skew $n$-shape} $\lambda/\mu$ is a skew shape with $n$ cells, and
we write $\lambda/\mu{\,\vdash\,} n$ or $|\lambda/\mu|=n$.
Here is an example of a skew 6-shape:
\begin{center}
  \setlength{\unitlength}{0.5mm}
  \begin{picture}(50,50)(0,-50)
    \put(-100,-27){$(6,4,2,2,1)/(4,3,2)\ \ =$}

    \dashline[-7]{1}(0,0)(60,0)
    \dashline[-7]{1}(0,-10)(60,-10)
    \dashline[-7]{1}(0,-20)(40,-20)
    \dashline[-7]{1}(0,-30)(20,-30)
    \dashline[-7]{1}(0,-40)(20,-40)
    \dashline[-7]{1}(0,-50)(10,-50)
    \dashline[-7]{1}(0,0)(0,-50)
    \dashline[-7]{1}(10,0)(10,-50)
    \dashline[-7]{1}(20,0)(20,-40)
    \dashline[-7]{1}(30,0)(30,-20)
    \dashline[-7]{1}(40,0)(40,-20)
    \dashline[-7]{1}(50,0)(50,-10)
    \dashline[-7]{1}(60,0)(60,-10)

    \put(40,0){\line(1,0){20}}
    \put(30,-10){\line(1,0){30}}
    \put(30,-20){\line(1,0){10}}
    \put(0,-30){\line(1,0){20}}
    \put(0,-40){\line(1,0){20}}
    \put(0,-50){\line(1,0){10}}
    \put(0,-30){\line(0,-1){20}}
    \put(10,-30){\line(0,-1){20}}
    \put(30,-10){\line(0,-1){10}}
    \put(20,-30){\line(0,-1){10}}
    \put(40,0){\line(0,-1){20}}
    \put(50,0){\line(0,-1){10}}
    \put(60,0){\line(0,-1){10}}
  \end{picture}
\end{center}

A {\em domino} is a rectangle consisting
of two cells. For an ordinary shape $\lambda$, let
$v(\lambda)$ denote the maximal number of disjoint
vertical dominoes that fit in the
shape $\lambda$.

A {\em (partial) tableau} $T$ on a skew $n$-shape
$\lambda/\mu$ is a labelling of the cells
of $\lambda/\mu$ with $n$ distinct real numbers
such that every number is greater
than its neighbours above and to the left.
We let $\sharp T=n$ denote the number of entries
in $T$, and ${\rm PT}(\lambda/\mu)$ denote the
set of partial tableaux on $\lambda/\mu$.

A {\em standard tableau} on a skew $n$-shape
is a tableau with the numbers $[n]=\{1,2,\ldots,n\}$.
We let ${\rm ST}(\lambda/\mu)$ denote the set of
standard tableaux on the shape $\lambda/\mu$.
Here is an example:
\begin{center}
  \setlength{\unitlength}{0.5mm}
  \begin{picture}(50,50)(0,-50)
    \dashline[-7]{1}(0,0)(60,0)
    \dashline[-7]{1}(0,-10)(60,-10)
    \dashline[-7]{1}(0,-20)(40,-20)
    \dashline[-7]{1}(0,-30)(20,-30)
    \dashline[-7]{1}(0,-40)(20,-40)
    \dashline[-7]{1}(0,-50)(10,-50)
    \dashline[-7]{1}(0,0)(0,-50)
    \dashline[-7]{1}(10,0)(10,-50)
    \dashline[-7]{1}(20,0)(20,-40)
    \dashline[-7]{1}(30,0)(30,-20)
    \dashline[-7]{1}(40,0)(40,-20)
    \dashline[-7]{1}(50,0)(50,-10)
    \dashline[-7]{1}(60,0)(60,-10)

    \put(40,0){\line(1,0){20}}
    \put(30,-10){\line(1,0){30}}
    \put(30,-20){\line(1,0){10}}
    \put(0,-30){\line(1,0){20}}
    \put(0,-40){\line(1,0){20}}
    \put(0,-50){\line(1,0){10}}
    \put(0,-30){\line(0,-1){20}}
    \put(10,-30){\line(0,-1){20}}
    \put(30,-10){\line(0,-1){10}}
    \put(20,-30){\line(0,-1){10}}
    \put(40,0){\line(0,-1){20}}
    \put(50,0){\line(0,-1){10}}
    \put(60,0){\line(0,-1){10}}
    \put(40,-10){\makebox(10,10){1}}
    \put(50,-10){\makebox(10,10){4}}
    \put(30,-20){\makebox(10,10){3}}
    \put(0,-40){\makebox(10,10){2}}
    \put(10,-40){\makebox(10,10){6}}
    \put(0,-50){\makebox(10,10){5}}
  \end{picture}
\end{center}
The (skew) shape of a tableau $T$ is denoted by $\sh T$.
Note that it is not sufficient to look at the cells of $T$
in order to determine its shape; we must think of the tableau
as remembering its underlying skew shape. (For instance,
$(6,4,2,2,1)/(4,3,2)$ and $(6,4,3,2,1)/(4,3,3)$ are distinct
skew shapes that have the same set of cells.)

The {\em sign} of a number sequence $w_1w_2\cdots w_k$
is $(-1)^{\sharp\{(i,j)\,:\,i<j,\,w_i>w_j\}}$, so it is
$+1$ for an even number of inversions, $-1$ otherwise.
The {\em inverse sign} is defined to be
$(-1)^{\sharp\{(i,j)\,:\,i<j,\,w_i<w_j\}}$.

The {\em sign} $\sgn T$ and the {\em inverse sign} $\invsgn T$
of a tableau $T$ are the sign respectively the inverse sign
of the sequence you get
by reading the entries row by row, from left to right and from
top to bottom, like a book.
Our example tableau has 4 inversions and 11 non-inversions,
so $\sgn T=+1$ and $\invsgn T=-1$.
\begin{defi}
  The {\em sign-imbalance} $I_{\lambda/\mu}$ of a skew shape
  $\lambda/\mu$ is the sum of the signs of all standard tableaux
  on that shape:
  $$I_{\lambda/\mu}=\sum_{T\in{\rm ST}(\lambda/\mu)}\sgn T.$$
\end{defi}
\noindent
An empty tableau has positive sign and
$I_{\lambda/\lambda}=I_\emptyset=1$.

A {\em biword} $\pi$ is a sequence of vertical pairs of positive
integers $\pi=\pair{i_1i_2\cdots i_k}{j_1j_2\cdots j_k}$
with $i_1\le i_2\le\cdots\le i_k$. We define the top and bottom
lines of $\pi$ by $\hat\pi=i_1i_2\cdots i_k$ and
$\check\pi=j_1j_2\cdots j_k$.
A {\em partial $n$-permutation} is a biword where in each line
the elements are distinct and of size at most $n$. Let
${\rm PS}_n$ denote the set of partial $n$-permutations.

For each $\pi\in{\rm PS}_n$ we associate an ordinary
$n$-permutation $\bar{\pi}\in S_n$
constructed as follows: First take the numbers among $1,2,\ldots,n$
that do not belong to $\hat\pi$ and sort them in increasing order
$a_1<a_2<\cdots<a_\ell$.
Then sort the numbers among $1,2,\ldots,n$
that do not belong to $\check\pi$ in increasing order
$b_1<b_2<\cdots<b_\ell$.
Now insert the vertical pairs $\pair{a_r}{b_r}$, $1\le r\le\ell$
into $\pi$ so that the top line remains increasingly ordered
(and hence must be $12\cdots n$). The bottom line is
a permutation (in single-row notation) which we denote $\bar\pi$.
Example: If $n=5$ and $\pi=\pair{124}{423}$ then $\bar\pi=42135$.

In the following we let ${\uplus}$ denote disjoint union interpreted
liberally. For instance, we will write $\check\pi{\uplus} T=[n]$
meaning that the set of numbers appearing in $\check\pi$ and the
set of entries of the tableau $T$ are disjoint and their union is $[n]$.

\section{The skew RS-correspondence\label{sec:skewrs}}
In~\cite{saganstanley} Bruce Sagan and Richard Stanley
introduced several analogues of the
Robinson-Schensted algorithm for skew Young tableaux. Their main
result is the following theorem.
\begin{theo}[Sagan and Stanley; 1990]\label{th:ss}
  Let $n$ be a fixed positive integer and $\alpha$ a fixed partition
  (not necessarily of $n$). Then there is bijection
  $$(\pi,T,U)\leftrightarrow(P,Q)$$
  between $\pi\in{\rm PS}_n$
  with $T,U\in{\rm PT}(\alpha/\mu)$ such that
  $\check\pi{\uplus} T=\hat\pi{\uplus} U=[n]$, on the one hand, and
  $P,Q\in{\rm ST}(\lambda/\alpha)$ such that $\lambda/\alpha{\,\vdash\,} n$,
  on the other.
\end{theo}
\noindent
Though we will assume detailed familiarity with it,
we do not define the bijection here, but refer to \cite{saganstanley}
for the original presentation.

\section{Our results\label{sec:results}}
\noindent
In~\cite{jonas} and \cite{astrid} the author and Astrid Reifegerste
independently discovered the formula for sign transfer under
the RS-correspondence:
\begin{theo}[Reifegerste; Sj\"ostrand; 2003]\label{th:jonasastrid}
  Under the (ordinary) RS-correspondence $\pi\leftrightarrow(P,Q)$ we have
  $$\sgn\pi=(-1)^{v(\lambda)}\sgn P\sgn Q$$
  where $\lambda$ is the shape of $P$ and $Q$.
\end{theo}
\noindent
Our main theorem is a generalization of this
to Sagan and Stanley's skew RS-correspondence:
\begin{theo}\label{th:main}
  Under the skew RS-correspondence $(\pi,T,U)\leftrightarrow(P,Q)$
  we have
  $$(-1)^{v(\lambda)}\sgn P\sgn Q=(-1)^{|\alpha|}(-1)^{v(\mu)+|\mu|}
  \sgn T\sgn U\sgn{\bar\pi}$$
  where $\sh P=\sh Q=\lambda/\alpha$ and $\sh T=\sh U=\alpha/\mu$.
\end{theo}
\noindent
Note that if
$\alpha=\emptyset$ the theorem reduces to Theorem~\ref{th:jonasastrid}.
\vspace{2mm}

\noindent
{\bf Remark.}
If we specialise to the skew RS-correspondence
$(\pi,T)\leftrightarrow P$ of involutions
(see Corollary~3.4 in~\cite{saganstanley}),
Theorem~\ref{th:main} gives that
$$(-1)^{v(\lambda)}=\sgn{\bar\pi}(-1)^{v(\mu)+|\mu|+|\alpha|},$$
where $\sh P=\lambda/\alpha$ and $\sh T=\alpha/\mu$.
This is also a simple consequence of Corollary~3.6
in~\cite{saganstanley} which is a generalization of a theorem by
Sch\"utzenberger \cite[page~127]{schutzenberger}
(see also \cite[exercise~7.28~a]{enum2}).
\vspace{2mm}

A fundamental application of Theorem~\ref{th:jonasastrid}
appearing in both \cite{jonas} and \cite{astrid} is the following
theorem.
\bigexample
\begin{theo}[Reifegerste; Sj\"ostrand; 2003]\label{th:jonasastrid2}
  For all $n\geq2$
  $$\sum_{\lambda\vdash n}(-1)^{v(\lambda)}I_\lambda^2=0.$$
\end{theo}
\noindent
We give a natural generalization of this using
Theorem~\ref{th:main}. It may be called a
``sign-imbalance analogue'' to Corollary~2.2 in~\cite{saganstanley}.
\begin{theo}\label{th:inout}
  Let $\alpha$ be a fixed partition and let $n$ be a positive integer.
  Then
  $$
  \sum_{\lambda/\alpha{\,\vdash\,} n}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2=
  \sum_{\alpha/\mu{\,\vdash\,} n}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2
  $$
  if $n$ is even, and
  $$
  \sum_{\lambda/\alpha{\,\vdash\,} n}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2=
  \sum_{\alpha/\mu{\,\vdash\,} n-1}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2
  -\sum_{\alpha/\mu{\,\vdash\,} n}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2
  $$
  if $n$ is odd.
\end{theo}
\noindent
Figure~\ref{fig:bigexample} gives an example.
Observe that if $\alpha=\emptyset$ and $n\ge2$ the theorem reduces to
Theorem~\ref{th:jonasastrid2}.

\section{The proof of the main theorem\label{sec:mainproof}}
For a skew shape $\lambda/\mu$, let
$$\rsgn{\lambda/\mu}:=(-1)^{\sum_{(r,c)\in\lambda/\mu}(r-1)}.$$
For convenience, let $\rsgn T:=\rsgn{\sh T}$ for a skew tableau $T$.
Observe that for an ordinary shape $\lambda$ we have
$\rsgn\lambda=(-1)^{v(\lambda)}$.

For the sake of bookkeeping we will make two minor adjustments to
the skew insertion algorithm that do not affect the resulting
tableaux:
\begin{itemize}
\item
  Instead of starting with an empty Q-tableau, we
  start with the tableau $U$ after multiplying all entries by $\varepsilon$.
  Here $\varepsilon$ is a very small positive number.
\item
  During an internal
  insertion a new cell with an integer $b$ is added to the Q-tableau
  according to the usual rules. New additional rule: At the same time
  we remove the entry $b\varepsilon$ from the Q-tableau.
\end{itemize}

Consider the (adjusted) skew insertion algorithm starting with
P-tableau $P_0=T$ and Q-tableau $Q_0=U\varepsilon$.
After $\ell$ insertions (external or internal) we have obtained the
tableaux $P_\ell$ and $Q_\ell$. The following two lemmas state what
happens when we make the next insertion.
\begin{lemma}\label{lm:external}
  Let $(P_{\ell+1},Q_{\ell+1})$ be the resulting tableaux after
  {\em external} insertion of the number $a_1$ into $(P_\ell,Q_\ell)$.
  Then
  $$\frac{\sgn{P_{\ell+1}}}{\sgn{P_\ell}}
  =\frac{\sgn{Q_{\ell+1}}}{\sgn{Q_\ell}}
  \frac{\rsgn{Q_{\ell+1}}}{\rsgn{Q_\ell}}(-1)^{\sharp Q_\ell}(-1)^m,$$
  where $m$ is the number of entries in $P_\ell$ that are less than $a_1$.
\end{lemma}
\begin{proof}
  We insert the number $a_1$ which pops a number $a_2$ at $(1,c_1)$
  which pops a number $a_3$ at $(2,c_2)$ and so on. Finally the number
  $a_r$ fills a new cell $(r,c_r)$,
  see Figure~\ref{fig:extinsert}.
  \begin{figure}[t]
    \begin{center}
      \setlength{\unitlength}{5mm}
      \begin{picture}(14,12)(0,0)
        \put(0,0){\resizebox{70mm}{!}{\includegraphics{fig2_extinsert}}}
        \put(14.4,9.4){$a_1$}
      \end{picture}
      \caption{External insertion of $a_1$. The shaded cells are
        counted by the sum
        $\sum_{i=2}^r(\beta_{i-1}-c_{i-1}+c_i-1-\gamma_i)$ in the proof.}
      \label{fig:extinsert}
    \end{center}

  \end{figure}

  For $2\le i\le r$, the relocation
  of $a_i$ multiplies the sign of the P-tableau by
  $(-1)^{\beta_{i-1}-c_{i-1}+c_i-1-\gamma_i}$,
  where $\sh{P_\ell}=\sh{Q_\ell}=\beta/\gamma$. Summation yields
  $$\sum_{i=2}^r(\beta_{i-1}-c_{i-1}+c_i-1-\gamma_i)
  =-(c_1-\gamma_1+r-2)+\sum_{i=1}^r(\beta_i-\gamma_i)$$
  since $\beta_r=c_r-1$.
  The placing of $a_1$ in the first row multiplies the sign of
  the P-tableau by $(-1)^{m-(c_1-1-\gamma_1)}$ where
  $m$ is the number of entries in $P_\ell$ that are
  less than $a_1$. We get
  $$\frac{\sgn{P_{\ell+1}}}{\sgn{P_\ell}}
  =(-1)^{m+1-r+\sum_{i=1}^r(\beta_i-\gamma_i)}.
  $$
  Obviously
  $$\frac{\invsgn{Q_{\ell+1}}}{\invsgn{Q_\ell}}
  =(-1)^{\sum_{i=1}^r(\beta_i-\gamma_i)}$$
  and
  $$\frac{\rsgn Q_{\ell+1}}{\rsgn Q_\ell}=(-1)^{r-1}.$$
  Since $\sgn R\invsgn R=(-1)^{\binom{\sharp R}{2}}$ for any
  tableau $R$, we have
  $$\frac{\invsgn Q_{\ell+1}}{\invsgn Q_\ell}
  =\frac{\sgn Q_{\ell+1}}{\sgn Q_\ell}(-1)^{\sharp Q_\ell}.$$
  Combining the equations above proves the lemma.
\end{proof}

\begin{lemma}\label{lm:internal}
  Let $(P_{\ell+1},Q_{\ell+1})$ be the resulting tableaux after
  {\em internal} insertion of the entry $a_1$ at $(r,c_0)$
  into $(P_\ell,Q_\ell)$. Then
  $$
  \frac{\sgn{P_{\ell+1}}}{\sgn{P_\ell}}
  =\frac{\sgn{Q_{\ell+1}}}{\sgn{Q_\ell}}
  \frac{\rsgn{Q_{\ell+1}}}{\rsgn{Q_\ell}}(-1)^{\sharp Q_\ell}.$$
\end{lemma}
\begin{proof}
  During an internal insertion the entry
  $a_1$ at $(r,c_0)$ pops a number $a_2$ at
  $(r+1,c_1)$ which pops a number $a_3$ at $(r+2,c_2)$
  and so on. Finally the number $a_k$ fills a new cell
  $(r+k,c_k)$, see Figure~\ref{fig:intinsert}.
  \begin{figure}[t]
    \begin{center}
      \setlength{\unitlength}{5mm}
      \begin{picture}(12,10)(0,0)
        \put(0,0){\resizebox{60mm}{!}{\includegraphics{fig3_intinsert}}}
        \put(6.2,7.3){$a_1$}
      \end{picture}
      \caption{Internal insertion starting with $a_1$. The shaded cells are
        counted by
        $\sum_{i=1}^k(\beta_{r+i-1}-c_{i-1}+c_{i}-1-\gamma_{r+i})$
        in the proof.}
      \label{fig:intinsert}
    \end{center}
  \end{figure}

  For $1\le i\le k$, the relocation of $a_i$ multiplies
  the sign of the P-tableau by
  $(-1)^{\beta_{r+i-1}-c_{i-1}+c_{i}-1-\gamma_{r+i}}$,
  where $\sh{P_\ell}=\sh{Q_\ell}=\beta/\gamma$. Summation yields
  $$\sum_{i=1}^k(\beta_{r+i-1}-c_{i-1}+c_{i}-1-\gamma_{r+i})
  =-k+\sum_{j=r}^{r+k}(\beta_j-\gamma_j)$$
  since $\beta_{r+k}=c_k-1$ and $\gamma_r=c_0-1$.

  What happens to the Q-tableau?
  According to our adjustments of the algorithm
  the entry $b\varepsilon$ at $(r,c_0)$ is removed and the entry $b$
  is added at the new cell at $(r+k,c_k)$. Observe that
  $b\varepsilon$ is the smallest element in $Q_\ell$;
  this is the very reason
  why we are making an internal insertion from its cell $(r,c_0)$.
  Also note that $b$ is the largest entry in $Q_{\ell+1}$. The transformation
  from $Q_\ell$ to $Q_{\ell+1}$ can be thought of as consisting of
  two steps: First we replace the entry $b\varepsilon$ by
  $b$, thereby changing the sign of the tableau by a factor
  $(-1)^{\sharp Q_\ell-1}$. Then we move the $b$ to the new cell at
  $(r+k,c_k)$, thereby changing the sign of the tableau by a factor
  $$(-1)^{-1+\sum_{j=r}^{r+k}(\beta_j-\gamma_j)}.$$
  Now, after observing that
  $$\frac{\rsgn{Q_{\ell+1}}}{\rsgn{Q_\ell}}=\frac{(-1)^{r+k}}{(-1)^r}
  =(-1)^k,$$
  the lemma follows.
\end{proof}
\vskip15mm
Now we are ready to prove our main theorem.
\begin{proof}[Proof of Theorem~\ref{th:main}.]
  From Lemma~\ref{lm:external} and~\ref{lm:internal}
  we deduce by induction that
  \begin{equation}\label{eq:intermed}
    \frac{\sgn P}{\sgn T}=\frac{\sgn Q}{\sgn U}\frac{\rsgn Q}{\rsgn U}
    (-1)^{\sum_{\ell=0}^{n-1}\sharp Q_\ell}(-1)^{\sum m}
  \end{equation}
  where $n=\sharp P$ and the last sum $\sum m$ is taken over
  all external insertions.

  Let $t_1<t_2<\cdots<t_g$ and
  $u_1<u_2<\cdots<u_g$ be the entries of $T$ and
  $U$, and write $\pi=\pair{i_1i_2\cdots i_h}{j_1j_2\cdots j_h}$.
  Let $\pi'$ be the permutation you get (in single-row notation)
  by preceding $\check\pi$ with
  the elements of $T$ decreasingly ordered,
  i.e.,~$\pi'=t_gt_{g-1}\cdots t_1j_1j_2\cdots j_h$.
  It is easy to see that
  the sum $\sum m$ equals the number of non-inversions of $\pi'$,
  i.e~pairs $i<j$ such that $\pi'(i)<\pi'(j)$. This means
  that $(-1)^{\sum m}=\invsgn{\pi'}$.

  What is the relationship between
  $\invsgn{\pi'}$ and $\sgn{\bar\pi}$?

  Let us go from $\pi'$ to $\bar\pi$ by a sequence
  of moves. Start with
  $$\pi'=t_gt_{g-1}\cdots t_1j_1j_2\cdots j_h.$$
  Move the first entry $t_g$ to position $u_g$:
  $$\underbrace{t_{g-1}t_{g-2}\cdots t_1\,j_\ast\cdots j_\ast\,t_g}
  _{\mbox{$u_g$ entries}}\,j_\ast\cdots j_\ast$$
  (Here the symbolic indices $\ast$ should be replaced by
  the sequence $1,2,\ldots,h$.)
  Next, move the entry $t_{g-1}$ to position $u_{g-1}$:
  $$\underbrace{t_{g-2}t_{g-3}\cdots t_1\,j_\ast\cdots j_\ast\,t_{g-1}}
  _{\mbox{$u_{g-1}$ entries}}\,j_\ast\cdots j_\ast\,t_g\,
  j_\ast\cdots j_\ast$$
  Continue until
  all elements of $T$ are moved. The resulting permutation is
  $\bar\pi$. After analysing what the moves do to the sign of the permutation,
  we obtain
  $$\sgn{\bar\pi}=(-1)^K\sgn\pi'$$
  where
  $$K=\sum_{i=1}^g(u_i-1).$$
  Note also that
  $$\invsgn\pi'=(-1)^{\binom{n}{2}}\sgn\pi'.$$

  Now look at
  $$\sum_{\ell=0}^{n-1}\sharp Q_\ell.$$
  If we define $K_\ell:=\sharp\{b\in U\ :\ \ell<b\}$
  we can write $\sharp Q_\ell=\ell+K_\ell$. Summation yields
  $$\sum_{\ell=0}^{n-1}\sharp Q_\ell
  =\sum_{\ell=0}^{n-1}(\ell+K_\ell)=\binom{n}{2}+K+\sharp U.$$

  Now we are ready to update~(\ref{eq:intermed}):
  $$\frac{\sgn P}{\sgn T}=\frac{\sgn Q}{\sgn U}\frac{\rsgn Q}{\rsgn U}
  \sgn{\bar\pi}(-1)^{\sharp U}.$$
  There remains only some cleaning-up. Observe that
  $$\frac{\rsgn Q}{\rsgn U}
  =\frac{\rsgn{\lambda/\alpha}}{\rsgn{\alpha/\mu}}
  =\rsgn\lambda\rsgn\mu=(-1)^{v(\lambda)}(-1)^{v(\mu)}$$
  and $\sharp U=|\alpha|-|\mu|$.
  This yields the result
  $$(-1)^{v(\lambda)}\sgn P\sgn Q=(-1)^{|\alpha|}(-1)^{v(\mu)+|\mu|}
  \sgn T\sgn U\sgn{\bar\pi}.$$
\end{proof}

\section{The proof of Theorem~\ref{th:inout}\label{sec:inoutproof}}
In Theorem~\ref{th:ss} we have adopted the original notation from
Sagan and Stanley \cite{saganstanley}. However, for some applications
(and among them the forthcoming proof of Theorem~\ref{th:inout})
it is inconvenient to work with {\em partial} tableaux. For that matter
we now present a simple bijection that will allow us to work with
{\em standard} tableaux only.
\begin{lemma}\label{lm:seq}
  Let $n$ be a fixed positive integer and $\alpha$ and $\mu$ fixed
  partitions.
  Then there is a bijection $(\pi,T,U)
  \leftrightarrow(\tilde\pi,\tilde I,\tilde T,\tilde U)$ between
  \begin{itemize}
  \item
    triples $(\pi,T,U)$
    such that $\pi\in{\rm PS}_n$, $T,U\in{\rm PT}(\alpha/\mu)$
    and $\check{\pi}{\uplus} T=\hat{\pi}{\uplus} U=[n]$, and
  \item
    quadruples $(\tilde\pi,\tilde I,\tilde T,\tilde U)$ such that
    $\tilde\pi\in S_n$,
    $\tilde T,\tilde U\in{\rm ST}(\alpha/\mu)$ and $\tilde I\subseteq[n]$
    is the index set of an increasing subsequence of
    $\tilde\pi$ of length $|\alpha/\mu|$.
  \end{itemize}
  This bijection has the following properties:
  \begin{align*}
    \tilde\pi & =\bar\pi, \\
    \sgn{\tilde T} & =\sgn T, \\
    \sgn{\tilde U} & =\sgn U.
  \end{align*}
\end{lemma}
\begin{proof}
  Given a quadruple $(\tilde\pi,\tilde I,\tilde T,\tilde U)$,
  let the triple $(\pi,T,U)$ be given by the following procedure:
  Write $\tilde\pi$ in biword notation and remove the vertical pairs
  corresponding to the increasing subsequence $\tilde I$. The resulting
  partial permutation is $\pi$. Order the elements in $\tilde I$
  increasingly: $i_1<i_2<\cdots<i_k$. Now, for $1\le j\le k$,
  replace the entry $j$ in $\tilde U$ by $i_j$ and
  replace the entry $j$ in $\tilde T$ by $\tilde\pi(i_j)$. This results
  in $U$ and $T$ respectively. It is easy to see that this is indeed a
  bijection with the claimed properties.
\end{proof}

Now we are ready to prove Theorem~\ref{th:inout}.
\begin{proof}[Proof of Theorem~\ref{th:inout}.]
  Sum the equation of Theorem~\ref{th:main} over
  the whole domain of the skew RS-correspondence according to
  Theorem~\ref{th:ss} in view of Lemma~\ref{lm:seq}:
  \begin{gather*}
    \sum_{\lambda/\alpha{\,\vdash\,} n}\;\sum_{P,Q\,\in\,{\rm ST}(\lambda/\alpha)}
    (-1)^{v(\lambda)}\sgn P\sgn Q=\\
    \sum_{k=0}^n\;\sum_{\alpha/\mu{\,\vdash\,} k}\;
    \sum_{T,U\in{\rm ST}(\alpha/\mu)}\,\,
    \sum_{1\le i_1<\cdots<i_k\le n}\!\!\!\!
    \sum_{\sarray{
        \pi\in S_n \\
        \pi(i_1)<\cdots<\pi(i_k)
      }}
    \!\!\!\!\!\!\!\!\!\!(-1)^{|\alpha|+v(\mu)+|\mu|}
    \sgn T\sgn U\sgn\pi
  \end{gather*}
  Let ${\rm LHS}$ and ${\rm RHS}$ denote the left-hand side and
  the right-hand side of the equation above.
  The left-hand side trivially equals
  $${\rm LHS}=\sum_{\lambda/\alpha{\,\vdash\,} n}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2.
  $$
  The right-hand side is trickier.
  Fix $1\le i_1<i_2<\cdots<i_k\le n$ and consider the sum
  $$S:=\sum_{\sarray{
      \pi\in S_n \\
      \pi(i_1)<\cdots<\pi(i_k)
    }}
  \!\!\!\!\!\!\sgn\pi.
  $$
  \begin{itemize}
  \item
    If $k=n$ clearly $S=1$.
  \item
    If $k\le n-2$ there are at least two integers
    $1\le a<b\le n$ not contained in the sequence
    $i_1<i_2<\cdots<i_k$. The sign-reversing involution
    $\pi\mapsto \pi\cdot(a,b)$ (here $(a,b)$ is
    the permutation that switches $a$ and $b$) shows that
    $S=0$.
  \item
    Suppose $k=n-1$ and let $a$ be the only integer in $[n]$
    not contained in the sequence $i_1<i_2<\cdots<i_k$.
    We are free to choose $\pi(a)$ from $[n]$, but as soon as
    $\pi(a)$ is chosen, the rest of $\pi$ must be the unique
    increasing sequence consisting of $[n]\setminus\pi(a)$
    if $\pi$ should contribute to $S$.
    The sign of $\pi$ then becomes $(-1)^{\pi(a)-a}$ so
    $$S=\sum_{i=1}^n(-1)^{i-a}=\left\{
      \begin{array}{ll}
        0 & \mbox{if $n$ is even,} \\
        (-1)^{a-1} & \mbox{if $n$ is odd.}
      \end{array}\right..$$
  \end{itemize}
  In the case where $n$ is odd and $k=n-1$, the double sum
  $$\sum_{1\le i_1<\cdots<i_k\le n}\!\!\!
  \sum_{\sarray{
      \pi\in S_n \\
      \pi(i_1)<\cdots<\pi(i_k)
    }}
  \!\!\!\!\!\!\sgn\pi
  =\sum_{a=1}^n(-1)^{a-1}=1.$$

  In summary we have showed
  $$\sum_{1\le i_1<\cdots<i_k\le n}\!\!\!
  \sum_{\sarray{
      \pi\in S_n \\
      \pi(i_1)<\cdots<\pi(i_k)
    }}
  \!\!\!\!\!\!\sgn\pi
  =\left\{\begin{array}{rl}
      1 & \mbox{if $k=n$,}\\
      1 & \mbox{if $k=n-1$ and $n$ is odd,}\\
      0 & \mbox{if $k=n-1$ and $n$ is even,}\\
      0 & \mbox{if $k\le n-2$.}
    \end{array}\right.$$

  If $n$ is even we finally obtain
  $${\rm RHS}=(-1)^{|\alpha|}\;\sum_{\alpha/\mu{\,\vdash\,} n}\;
  (-1)^{v(\mu)+|\mu|}
  \sum_{T,U\in{\rm ST}(\alpha/\mu)}
  \sgn T\sgn U
  =\sum_{\alpha/\mu{\,\vdash\,} n}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2
  $$
  since $(-1)^{|\alpha|+|\mu|}=(-1)^{|\alpha|-|\mu|}=(-1)^n=1$.

  Analogously, if $n$ is odd we get
  \begin{align*}
    {\rm RHS} & =(-1)^{|\alpha|}\;\sum_{\alpha/\mu{\,\vdash\,} n}\;
    (-1)^{v(\mu)+(|\alpha|-n)}
    \sum_{T,U\in{\rm ST}(\alpha/\mu)}
    \sgn T\sgn U\\
    & \quad\mbox{}+(-1)^{|\alpha|}\;\sum_{\alpha/\mu{\,\vdash\,} n-1}\;
    (-1)^{v(\mu)+(|\alpha|-(n-1))}
    \sum_{T,U\in{\rm ST}(\alpha/\mu)}
    \sgn T\sgn U\\
    & =\sum_{\alpha/\mu{\,\vdash\,} n-1}
    (-1)^{v(\mu)}I_{\alpha/\mu}^2
    -\sum_{\alpha/\mu{\,\vdash\,} n}
    (-1)^{v(\mu)}I_{\alpha/\mu}^2.
  \end{align*}
\end{proof}

\section{Specialisations of Theorem~\ref{th:inout}\label{sec:specialisations}}
Apart from the special case $\alpha=\emptyset$,
Theorem~\ref{th:inout} offers a couple of other nice
specialisations if we choose the parameters $\alpha$ and $n$ properly.
First we obtain a surprising formula for the square of
the sign-imbalance of any ordinary shape:
\begin{coro}
  Let $\alpha$ be a fixed $n$-shape. Then
  $$I_\alpha^2
  =\sum_{\lambda/\alpha{\,\vdash\,} n}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2
  =\sum_{\lambda/\alpha{\,\vdash\,} n+1}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2$$
  if $n$ is even, and
  $$I_\alpha^2
  =\sum_{\lambda/\alpha{\,\vdash\,} n-1}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2
  $$
  if $n$ is odd.
\end{coro}
\begin{proof}
  First suppose $n$ is even.
  Theorem~\ref{th:inout} yields
  $$
  \sum_{\lambda/\alpha{\,\vdash\,} n}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2=
  \sum_{\alpha/\mu{\,\vdash\,} n}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2.
  $$
  The right-hand side consists of only one term, namely
  $(-1)^{v(\emptyset)}I_{\alpha/\emptyset}^2=I_\alpha^2$.
  From Theorem~\ref{th:inout} we also get
  $$
  \sum_{\lambda/\alpha{\,\vdash\,} n+1}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2=
  \sum_{\alpha/\mu{\,\vdash\,} n}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2
  -\sum_{\alpha/\mu{\,\vdash\,} n+1}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2.
  $$
  The second term of the right-hand side vanishes and the first term
  is $I_\alpha^2$ as before.

  Now suppose $n$ is odd. Then Theorem~\ref{th:inout} yields
  $$
  \sum_{\lambda/\alpha{\,\vdash\,} n-1}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2=
  \sum_{\alpha/\mu{\,\vdash\,} n-1}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2
  $$
  The right-hand side consists of only one term, namely
  $(-1)^{v((1))}I_{\alpha/(1)}^2$ which equals $I_\alpha^2$
  since in an ordinary tableau the 1 is always located at $(1,1)$.
\end{proof}

Next we present another generalization of Theorem~\ref{th:jonasastrid2}.
\begin{coro}
  Let $\alpha$ be a fixed $n$-shape. Then
  $$\sum_{\lambda/\alpha{\,\vdash\,} m}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2=0$$
  for any integer $m\ge n+2$ if $n$ is even, and for
  any integer $m\ge n$ if $n$ is odd.
\end{coro}
\begin{proof}
  If $m$ is even Theorem~\ref{th:inout} yields
  $$
  \sum_{\lambda/\alpha{\,\vdash\,} m}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2=
  \sum_{\alpha/\mu{\,\vdash\,} m}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2.
  $$
  The right-hand side vanishes since $m>|\alpha|$.

  If $m$ is odd Theorem~\ref{th:inout} yields
  $$
  \sum_{\lambda/\alpha{\,\vdash\,} m}
  (-1)^{v(\lambda)}I_{\lambda/\alpha}^2=
  \sum_{\alpha/\mu{\,\vdash\,} m-1}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2
  -\sum_{\alpha/\mu{\,\vdash\,} m}
  (-1)^{v(\mu)}I_{\alpha/\mu}^2.
  $$
  If $m\ge n+2$ the right-hand side vanishes simply because
  $m-1>|\alpha|$. Otherwise $n$ is odd and the only remaining case
  is $m=n$. But then the right-hand side becomes
  $I_{\alpha/(1)}^2-I_\alpha^2=0$.
\end{proof}

\section{Future research\label{sec:future}}
For an ordinary shape $\lambda$, let $h(\lambda)$ be the
number of disjoint horizontal dominoes that fit in $\lambda$
and let $d(\lambda)$ be the number of disjoint
$2\times2$-squares (fourlings) that fit in $\lambda$.

In~\cite{jonas} the following theorem, conjectured
by Stanley \cite{stanley},
was proved (the (a)-part was independently proved by T.~Lam~\cite{lam}):
\begin{theo}[Stanley; Lam; Sj\"ostrand; 2003]\label{th:stanleyab}
\mbox{\\}
\begin{enumerate}
\item[(a)] For every $n\geq0$
$$\sum_{\lambda\vdash n}q^{v(\lambda)}t^{d(\lambda)}x^{h(\lambda)}
I_\lambda=(q+x)^{\lfloor n/2\rfloor}.$$
\item[(b)] If $n\not\equiv1\pmod4$
$$\sum_{\lambda\vdash n}(-1)^{v(\lambda)}t^{d(\lambda)}I_\lambda^2=0.$$
\end{enumerate}
\end{theo}
The (b)-part is a strengthening of Theorem~\ref{th:jonasastrid2}
and one might wonder if there is a similar strengthening
of Theorem~\ref{th:inout} for skew shapes.

The (a)-part
is about signed sums of sign-imbalances without taking
the square. From an RS-correspondence perspective
it is unnatural not to take the square of the sign-imbalance
since the P- and Q-tableaux come in pairs. In fact it
might be argued that non-squared sign-imbalances are unnatural
in all cases, because their sign is dependent on the actual
labelling of the poset, i.e.,~it is important that
we read the tableau as a book. However, part (a)
in the theorem is still true (and there are even stronger theorems,
see~\cite{jonas}) and it can be proved by means of the
RS-correspondence as was done in~\cite{jonas}. This suggests
that the skew RS-algorithm could be a useful tool for
studying signed sums of non-squared sign-imbalances too.

As a tool for proving Theorem~\ref{th:stanleyab} the concept of
{\em chess tableaux} was introduced in~\cite{jonas}. A chess tableaux
is a standard Young tableau where odd entries are located at an
even Manhattan distance from the upper-left cell of the shape,
while even entries are located at odd distances.
This notion of course generalizes to skew tableaux
(in fact it generalizes to many
other posets) and since it proved so useful in the study of
sign-imbalance of ordinary shapes we think it will shed some light
on the skew shapes as well.

Another direction of research is to find analogues to
Theorem~\ref{th:main} for other variants of the RS-algorithm.
For instance, in~\cite[Theorem~5.1]{saganstanley}
Sagan and Stanley present a
generalization of their skew RS-correspondence where
the condition that $\sh U=\sh T$ and $\sh P=\sh Q$ is
relaxed. From that they are able to infer identities like
$$\sum_\sarray{
\lambda/\beta{\,\vdash\,} n \\
\lambda/\alpha{\,\vdash\,} m
}
f_{\lambda/\beta}f_{\lambda/\alpha}
=\sum_{k\ge0}\binom{n}{k}\binom{m}{k}k!
\sum_\sarray{
\alpha/\mu{\,\vdash\,} n-k \\
\beta/\mu{\,\vdash\,} m-k
}
f_{\alpha/\mu}f_{\beta/\mu}$$
where $f_{\lambda/\mu}=\sharp{\rm ST}(\lambda/\mu)$. This correspondence
may give interesting formulas for sums of products of
sign-imbalances as well.

\section*{Acknowledgements}
I want to thank Richard Stanley and Frank Sottile for
personal communication.


text:
\section{Introduction}\label{intro}
Over the last two decades, far-infrared (FIR) and millimeter observations of high-redshift quasars ($z > 6$) have produced estimates for galactic dust masses as high as $10^{8}~M_{\odot}$ \citep{Smail:1997gf, Hughes:1998ve, Bertoldi:2003fr, Wang:2008rt}.  In order to explain the formation of this large quantity of dust within the short lifetime ($\sim$1 Gyr) of the universe at this epoch, a mechanism must exist which is capable of both significant and rapid dust production.  One recently pursued solution is that the majority, if not all, of this dust comes from core-collapse supernova (CCSN) explosions of the first generations of stars \citep{Morgan:2003lr, Maiolino:2004bh, Hirashita:2005vn}.

While various theoretical work \citep{Kozasa:1989ve, Kozasa:1991ul, Todini:2001qf, Nozawa:2003pd, Bianchi:2007fp, Nozawa:2010yq} has suggested that $\sim$0.1-0.3~$M_{\odot}$ of dust could be formed per supernova event, which is in rough agreement with the $\sim$0.2-1.0~$M_{\odot}$ required to explain high-redshift dust \citep{Morgan:2003lr, Dwek:2007lr, Dwek:2011fj, Gall:2011uq}. However, observational efforts focused on local supernova remnants (SNRs) often fall orders of magnitude short of these values \citep{Stanimirovic:2005rr, Williams:2006ly, Meikle:2007dq, Rho:2008qf, Rho:2009ul, Kotak:2009lr}.  More recent observations aimed at finding colder dust ($T < 40$~K) using AKARI and the Balloon-borne Large Aperature Submillimeter Telescope (BLAST) by \cite{Sibthorpe:2010fk}, as well as {\it Herschel} by \cite{Barlow:2010lr} and \cite{Matsuura:2011qy}, find larger dust masses.  \cite{Matsuura:2011qy} find $\sim$0.6~$M\subscript{\odot}$ of dust in SN1987A.  Such estimates could make the argument for SNe as dust factories more plausible.

One question that arises from these SNR studies is what fraction of the freshly formed dust predicted by theory will survive the interaction between the reverse shocks and the ejecta.  As the fast-moving dust ($V_{ej} \ge 1000$~km~s$^{-1}$) is impacted by the reverse shock, it will be subject to sputtering and grain-grain collisions as the density and temperature of the dust-enriched gas are increased.

In our previous work \citep[][hereafter Paper I]{Silvia:2010lr}, we investigated, through hydrodynamic simulations, the destruction of newly formed dust grains by sputtering in the reverse shocks of SNRs. Using ``cloud-crushing" simulations \citep{Woodward:1976pd, Mac-low:1994fj, Klein:1994uq}, we found that the degree of dust destruction depends heavily on the initial radius distribution of the dust grains as well as the initial density of the ejecta cloud and relative velocity between the reverse shock and the cloud.  In the most extreme cases, we found grain destruction to vary from 20-100\% depending on grain species.  We also found morphological similarities within these simulations to the observational studies of Cassiopeia A presented in \cite{Fesen:2011kx}, specifically in the scenarios where the effects of cooling led to the fragmentation of ejecta clouds.

However, our earlier simulations computed the radiative cooling and sputtering rates for approximately solar metallicity ($Z_{\odot}$) gas.  In reality, the gas contained in supernova ejecta will be highly metal-enriched. The question arises as to the balance between the enhanced radiative cooling and the increase in sputtering yields in this high-metallicity regime.  This balance will also be influenced by the changing density of the metal-enriched gas as the cloud is shredded and the plasma thins out.

In the current paper, we perform additional cloud-crushing simulations to study the evolution of dust mass contained within an idealized ejecta knot as it is impacted by a supernova reverse shock.  In Section \ref{methods} we give a brief review of the code used to carry out these simulations, the methods for tracking our dust populations, and the changes made from our previous work.  In Section \ref{sims}, we describe the simulations unique to this paper, specifically aimed at probing the higher metallicity regimes expected to be present in supernova ejecta.  We present the results of these simulations in Section \ref{results} and conclude in Section \ref{summary} with a summary and discussion of the implications of these results.

\section{Methodology}\label{methods}

\subsection{Code and Simulation Setup}\label{code}

As in Paper I, we use the Eulerian adaptive mesh refinement (AMR), hydrodynamics + N-body code, \texttt{Enzo} \citep{Bryan:1997qy, Norman:1999uq, Oshea:2004book}.  We do not make use of any of the cosmological or gravity-solving components of the code owing to the idealized nature of our problem.  We follow the same cloud-crushing setup as outlined in Paper I and refer the reader to that work for a detailed description.

The user-supplied parameters required to initialize the simulation are: cloud radius, $r\subscript{cloud}$; cloud temperature,  $T\subscript{cloud}$; density of the ambient medium, $\rho\subscript{m}$; initial over-density, $\chi$, of the cloud with respect to the ambient medium ($\rho\subscript{cloud} = \chi\rho\subscript{m}$); and the velocity of the shock relative to the stationary cloud, $v\subscript{shock}$.  From these input parameters, the following values must be derived in order to completely initialize the simulation: temperature of the ambient medium, $T\subscript{m}$, post-shock density, $\rho\subscript{shock}$, and post-shock temperature, $T\subscript{shock}$.  We set $T\subscript{m}$ so that the cloud remains in pressure equilibrium, while the shock-related values are calculated using the Rankine-Hugoniot jump conditions.

During runtime, we take advantage of the AMR capabilities of \texttt{Enzo} by employing the same refinement criterion as described in Paper I, increasing resolution in areas of the simulation that contain a significant fraction of cloud material.  Cells that are initially enclosed within the cloud radius are assigned a ``cloud material" value, $\xi$, that is advected with the flow in the same way as density.  When a cell exceeds a pre-defined cloud material ``mass", $m\subscript{\xi} = \xi V\subscript{cell}$, where $V\subscript{cell}$ is the cell volume, the resolution of that cell is doubled.  A more thorough description of this refinement process can be found in Paper I.

\subsection{Dust Tracking and Post-processing}\label{dusttrack}
Our dust-grain populations continue to be tracked through tracer particles embedded in the flowing gas.  As before, we post-process the density and temperature histories of each tracer particle to compute dust survival rates.  For the initial distributions in grain radii, we again made use of the values calculated by \cite{Nozawa:2003pd} for a CCSN with a progenitor mass of 20~$M\subscript{\odot}$ (see Figure \ref{rad_dist}).  We follow the evolution of all nine grain species included in the unmixed grain model of \cite{Nozawa:2003pd}.  To determine the evolution in dust mass, we use the same mass proxy outlined in Paper I, which involves tracking the changes in $n(a) \times a^{3}$, where $n(a)$ is the number density of grains and $a$ is the grain radius.  However, the erosion rates used to sputter the grains with each successive time step are different than those used in Paper~I, as described in the following section.  We note that we only account for dust grain destruction in this work and do not include a prescription for possible grain growth that might occur at high densities and low temperatures.  For the majority of our simulations, the amount of time that the dust grains spend in an environment conducive to grain growth is relatively brief and any potential increase in dust mass should be minimal.

\begin{figure}[htp]
\centering
  		\includegraphics[width=0.5\textwidth]{sizedist_unmixed.eps}
  \caption{Reproduced grain radius distributions for the species expected in the unmixed ejecta model of a core-collapse supernova with a progenitor mass of 20 M$\subscript{\odot}$ as calculated by \cite{Nozawa:2003pd}. The $y$-axis is the abundance of grains of a given radius, $f(a)$, such that the number density of grains between $a$ and $a + \delta a$ is $n(a) = f(a) \delta a$, where $\delta a$ is set by the number of bins used to track the distribution.}\label{rad_dist}
\end{figure}

\subsection{Modifications and Additions}\label{modifications}

To account for the fact that we expect the supernova ejecta clumps to be metal-enriched, we implement both new cooling curves and grain erosion rates for high-metallicity gas.  We calculate the cooling rates using \texttt{Cloudy} \citep{Ferland:1998sf} for gas assumed to be in ionization equilibrium.  In addition, these rates assume that both the electrons and the different ion species have the same temperature.  Cooling due to the thermal emission of dust grains is not included.

For this work, we compute rates for metallicities of $Z/Z\subscript{\odot} =$ 1, 10, and 100.  The element abundances for solar metallicity gas come from the composition listed in the documentation for \texttt{Cloudy}, where all abundances are specified by number relative to hydrogen.  We define the metallicities with values greater than unity to mean that the abundances for metals are increased by factors of 10 and 100 from their solar values.  The abundance of hydrogen and helium remain the same for all three metallicities.  To simplify the notation, we refer to these metallicities as 1~$Z\subscript{\odot}$, 10~$Z\subscript{\odot}$, and 100~$Z\subscript{\odot}$ for the remainder of the paper.  We also note that the values for the metal abundances remain static for the duration of our simulations, and any potential increase in metal abundance due to sputtered dust grains is omitted.  We comment on this omission in the final section of the paper.  Figure \ref{cool_erode}a shows the cooling rate coefficients for 1, 10, and 100~$Z\subscript{\odot}$ gas as a function of temperature.  To understand the contributions that hydrogen, helium, and heavier elements make on the cooling curves, we refer the reader to \cite{Gnat:2007fk}.

In contrast to Paper I, in which the cooling rates were applied to all cells within the simulation domain, we only cool those cells that contain cloud material.  This reduces the computational time required to calculate the cooling rates and accounts for the fact that, while the cloud itself can be highly metal-enriched, the ambient medium is at much lower densities and metallicities.  To ensure that important information is not lost by cooling only the cloud, we consider that for an ambient medium with a metallicity of $Z = 1~Z\subscript{\odot}$, a number density of $n = 1$~cm$^{-3}$, and a temperature of $T = 10^6$~K, the cooling time is a few~$\times~10^4$~yrs, much longer than the time scale for any of the simulations included in this work.

\begin{figure}[htp]
\centering
	\subfigure[Metallicity-dependent cooling curves]{
  		\includegraphics[width=0.465\textwidth]{coolingcurves.eps}
	}
  	\subfigure[Metallicity-dependent erosion rates]{
		\includegraphics[width=0.465\textwidth]{erosion_rates_3species.eps}
	}
  \caption{(a) Radiative cooling curves for solar-scaled metal abundances $Z = 1~Z\subscript{\odot}$, 10~$Z\subscript{\odot}$, and 100~$Z\subscript{\odot}$. (b) Dust grain erosion rates for C, SiO$\subscript{2}$, and Fe grains.}\label{cool_erode}
\end{figure}

We calculate new erosion rates using the formula provided by \cite{Nozawa:2006ve} for the limit in which thermal sputtering is the dominant mechanism for grain erosion.  This is appropriate, since the tracer particles used to track the dust are embedded in the flow.  In computing these erosion rates, we use the same scaled solar metal abundance ratios that were used to produce our new cooling functions.  Computing rates based on these enhanced metal abundances is an important step.  The sputtering yield, $Y(E)$ where $E$ is the energy of the impacting ion, is strongly dependent on the atomic mass of the impactor; high-mass ions can have orders of magnitude higher yields at high energies.  The differences in erosion rates as a function of metallicity are shown for carbonaceous, silicate, and ferrous grains in Figure \ref{cool_erode}b.  The differences do not become significant until the temperature exceeds a few times $10^{6}$~K and, even then, only for the highest metallicity.  In order to simplify Figure \ref{cool_erode}b, we have omitted the other six grain species studied in this work.  However, the erosion rates for the omitted species are comparable to the three presented species.

We include Figure \ref{sput_yield}a to show the nature of the sputtering yields for a sample of impacting ions as a function of energy and Figure \ref{sput_yield}b to show the contribution of various impacting ions to overall grain sputtering for gas with both a metallicity of $1~Z\subscript{\odot}$ and $100~Z\subscript{\odot}$.  For simplicity, we only present these quantities for C grains, but note that the figures are qualitatively similar for the other grain species.  From Figure \ref{sput_yield}b, it is evident that the contributions from the sputtering yields of high-mass ions do not begin to dominate over hydrogen and helium until the metallicity reaches $100~Z\subscript{\odot}$, which agrees with the trend observed in the erosion rates presented in Figure \ref{cool_erode}b.

\begin{figure}[htp]
\centering
	\subfigure[Sputtering yields]{
  		\includegraphics[width=0.465\textwidth]{YieldPlot_CloudyZ1_C.eps}
	}
  	\subfigure[Metallicity-dependent contributions to sputtering]{
		\includegraphics[width=0.465\textwidth]{QPlot_CloudyZ1_C.eps}
	}
  \caption{(a) Sputtering yields, $Y\subscript{i}(E)$, of C grains versus projectile energy, E, for various incident ions; (b) The contribution of various incident ions to the sputtering of dust grains, $Q\subscript{i}$, where $Q\subscript{i}$ is defined to be the product of the element abundance by number relative to hydrogen, $n\subscript{i}/n\subscript{H}$, and the sputtering yield normalized to that of hydrogen, $Y\subscript{i}/Y\subscript{H}$, as a function of energy.  We show the contributions based on solar abundance ratios (solid lines) and abundance ratios for $Z = 100~Z\subscript{\odot}$ (dashed lines).}\label{sput_yield}
\end{figure}

When these new cooling and sputtering rates are implemented within our simulations, we find that, for the high metallicity ($Z = 100~Z_{\odot}$) simulations, the densest cells end up with extremely short cooling times, often shorter than the time step set by the Courant-Friedrichs-Lewy (CFL) condition.  This tends to result in cells that over-cool during a given time step and can lead to negative cell energies.  To avoid this issue, we make two modifications to \texttt{Enzo}.  First, we add a temperature floor, $T\subscript{floor} = 1000$~K, to all of our simulations to prevent clouds from becoming unrealistically cold.  Given the energetic environment of the supernova remnant and the background radiation from shock-heated, x-ray emitting gas in the remnant's shell, such a floor is physically reasonable.  Second, we modify the time-step calculation such that it is never more than 25\% of the cooling time.  While this prevents negative cell energies, it can become computationally expensive, as the cloud is compressed to high densities and the cooling time becomes very short in some regions of the cloud.  It is this computational limitation that prevents us from exploring even higher metallicities, at least within the bounds of our computational resources.

\section{Simulations}\label{sims}
All of the simulations presented in this work have root-grid dimensions of 512~$\times$~256~$\times$~256 and a physical resolution of 1.25~$\times~10^{15}$~cm ($\sim$84~AU) per cell edge.  As in our previous work, we initialize each cloud to have a radius of $r\subscript{cloud}=10^{16}$~cm, which spans 8 cells at the root-grid resolution.  We also use the same Gaussian envelope formulation as in Paper I, with the density fall-off occurring at $r = 0.7~r\subscript{cloud}$.

We allow the simulation domain to be refined up to three additional levels such that the highest resolution cells will be 8 times smaller than the root-grid.  This differs from our previous work, where we only allowed for two additional levels of resolution.  The third level of refinement was required specifically in the simulations with high metallicity, to follow the collapse of some fragments to much higher densities and smaller spatial scales than in the low-metallicity cases.  When we allow for a fourth level of refinement, the final dust masses do not change significantly; we therefore limit ourselves to three levels to save computational resources. As mentioned above, our cell refinement is based on the amount of cloud material in a given cell.

For this work, we primarily focus our exploration of parameter space to the relative velocity between the reverse shock and the ejecta cloud and the metallicity of the gas contained within the cloud.  Specifically, we investigate shock velocities of 10$^{3}$~km~s$^{-1}$, 3$\times$10$^{3}$~km~s$^{-1}$, 5$\times$10$^{3}$~km~s$^{-1}$, and 10$^{4}$~km~s$^{-1}$ and metallicities of 1~$Z\subscript{\odot}$, 10~$Z\subscript{\odot}$, and 100~$Z\subscript{\odot}$. For ease of reference, we provide a numbered list of all simulations in Table \ref{sim_table}.  For all simulations, the cloud constrast is $\chi = 1000$, the initial cloud temperature is set equal to our temperature floor, $T\subscript{cloud} = T\subscript{floor} = 1000$~K, and the total runtime is $4.2t\subscript{cc}$, where $t\subscript{cc} = \chi^{1/2}r\subscript{cloud}/v\subscript{shock}$ is the cloud-crushing time \citep{Klein:1994uq}.

\begin{deluxetable}{lrrrrr}
\tabletypesize{\scriptsize}
\tablecaption{Simulation Parameters \label{sim_table}}
\tablewidth{0pt}
\tablehead{\colhead{{\bf Simulation}} &
	\colhead{$\chi$} &
	\colhead{$v\subscript{shock}$} &
	\colhead{$t\subscript{cc}$} &
	\colhead{Metallicity} \\
	\colhead{} &
	\colhead{} &
	\colhead{(km s$^{-1}$)} &
	\colhead{(yrs)} &
	\colhead{($Z/Z_{\odot}$)}}
\startdata
{\bf 1}	&	1000		&	1000		&	100.2	&	1	\\
{\bf 2}	&	1000		&	1000		&	100.2	&	10	\\
{\bf 3}	&	1000		&	1000		&	100.2	&	100	\\
{\bf 4}	&	1000		&	3000		&	33.4		&	1	\\
{\bf 5}	&	1000		&	3000		&	33.4		&	10	\\
{\bf 6}	&	1000		&	3000		&	33.4		&	100	\\
{\bf 7}	&	1000		&	5000		&	20.0		&	1	\\
{\bf 8}	&	1000		&	5000		&	20.0		&	10	\\
{\bf 9}	&	1000		&	5000		&	20.0		&	100	\\
{\bf 10}	&	1000		&	10000	&	10.0		&	1	\\
{\bf 11}	&	1000		&	10000	&	10.0		&	10	\\
{\bf 12}	&	1000		&	10000	&	10.0		&	100
\enddata
\tablecomments{Parameter definitions: $\chi$ is the initial over-density of the cloud with respect to the ambient medium, $v\subscript{shock}$ is the relative velocity between the inflowing shock and the stationary cloud, and $t\subscript{cc}$ is the cloud-crushing time.}
\vspace{5mm}
\end{deluxetable}

Figure \ref{simulation_evol} shows the evolution in density of the shock-cloud interaction from the initial impact to $t \sim 3~t\subscript{cc}$ for $v\subscript{shock} = 5 \times 10^3$ km~s$^{-1}$ and $Z = 100~Z\subscript{\odot}$.  Notable features include the tail of ablated material produced as the shock washes over the cloud, and the numerous cold, dense fragments that form a result of the high cooling rates for the metal-enriched gas.  A handful of these fragments persist for a large fraction of the total simulation time.

\begin{figure*}[htp]
\centering
	\includegraphics[width=1.0\textwidth]{multiPanelProjection_9_Density_Dust.pdf}
	\caption{Density projections in the $x$-$z$ plane weighted by cloud material for the case of $v\subscript{shock} = 5000$~km~s$^{-1}$ and $Z = 100~Z\subscript{\odot}$.  The number in the lower left corner of each panel indicates the time of the snapshot in units of the cloud-crushing time, $t\subscript{cc}$.  Projections were made using the software analysis package, \texttt{yt} \citep[yt-project.org;][]{Turk:2011lr}.}\label{simulation_evol}
\end{figure*}

While our previous work ran all simulations for a total of ten cloud-crushing times, we found that in our new Simulations 2 and 3 (slow shock and high metallicities) it was difficult to keep the majority of the dust particles within the computational domain much beyond four cloud-crushing times without significantly increasing the size of the root grid.  Though possible, we deemed this solution to be a poor use of our computational resources.  As will be seen in Section \ref{results}, the dust masses are still evolving in the relatively slow-shock ($v\subscript{shock} =  10^{3}$~km~s$^{-1}$ and 3$\times$10$^{3}$~km~s$^{-1}$) simulations at the point of termination.  However, the fast-shock simulations do not have significant changes in dust mass beyond that time.  Therefore, while we do see clear trends in the slow-shock simulations as a function of metallicity, we cannot say definitively how these trends would behave at later times.

\section{Results}\label{results}
We present the results for the twelve simulations listed in Table \ref{sim_table}, with a focus on the evolution in total dust mass and the time spent by the post-processed dust particles in various areas of density-temperature phase-space.

\subsection{Dust Mass Evolution}
The evolution of dust mass for each of the nine dust species and for all twelve simulations can be seen in Figure \ref{dustmassevol}, and the final dust masses can be located in Table \ref{dustsurvival}.  The most notable difference between the simulations is the drastic increase in dust destruction between the slowest shock velocity, $v\subscript{shock} = 10^{3}$~km~s$^{-1}$, and the highest shock velocity, $v\subscript{shock} = 10^{4}$~km~s$^{-1}$, in some cases a difference of $>$70\%.  For the intermediate shock velocities, $v\subscript{shock} = 3 \times 10^{3}$~km~s$^{-1}$ and $5 \times 10^{3}$~km~s$^{-1}$, the evolution in dust mass depends on the grain species.  For many species, the simulations with $v\subscript{shock} = 5 \times 10^{3}$~km~s$^{-1}$ show comparable dust destruction to the highest velocity shocks, while the simulations with $v\subscript{shock} = 3 \times 10^{3}$~km~s$^{-1}$ exhibit only moderate dust destruction.  There does appear to be a fundamental difference between the simulations with the slowest shock velocities and those with faster shocks at early times.  For nearly all grain species, there is a sharp drop in dust mass within one cloud-crushing time for shock velocities of $3 \times 10^{3}$~km~s$^{-1}$ and higher, with the magnitude of the drop increasing with shock velocity.  This suggests that there is a threshold velocity somewhere between $10^{3}$ and $3 \times 10^{3}$~km~s$^{-1}$ at which the initial impact of the shock into the cloud results in a degree of shock heating and compression that significantly influences the overall dust destruction in the cloud.  For slower shocks, the majority of the dust destruction occurs at much later times, as the cloud is shredded and incorporated into in the hot, post-shock gas.

In our study of the effects of metallicity for a given shock velocity, we find that only in the slowest shocks does an increase in metallicity consistently lead to a decrease in dust destruction (see the results of Simulation 3).  In the case of $v\subscript{shock} = 3 \times 10^{3}$~km~s$^{-1}$ and $Z = 100~Z\subscript{\odot}$, the amount of dust destruction is noticeably less for some grain species (Al$_2$O$_3$, MgSiO$_3$) and considerably more for others (Fe, Si).  This suggests that, for some grain species, the increased cooling due to the enhanced metallicity is able to counter-balance the shock-heating and keep the gas in a regime of decreased sputtering.  For the grain species with lower survival rates, a shock velocity of $3 \times 10^{3}$~km~s$^{-1}$ is sufficient to drive up the amount of dust destruction.  However, we again note that this is only based on the results at $t = 4.2t\subscript{cc}$.  At this velocity, the simulations with $Z = 1~Z\subscript{\odot}$ and $10~Z\subscript{\odot}$ appear to have hit a dust mass plateau, but for $Z = 100~Z\subscript{\odot}$ the evolution in dust mass still has a clear downward slope.  For $v\subscript{shock} = 5 \times 10^{3}$~km~s$^{-1}$ and $10^{4}$~km~s$^{-1}$, an increase in metallicity leads directly to an increase in total dust destruction.  The most drastic change occurs when the metallicity is increased from $Z = 10~Z\subscript{\odot}$ to $100~Z\subscript{\odot}$; $\sim$20\% more of the initial dust is lost with this change in metallicity for Fe and Si grains in the highest velocity scenario.  For all simulations in which the evolution in dust mass plateaus at late times, the cloud has been significantly shredded and the density of the gas has been reduced to a level that does not allow for appreciable grain sputtering.

\begin{figure*}[htp]
\centering
	\subfigure[Al$_2$O$_3$]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_Al2O3.eps}
  	}
	\hspace{-5.00mm}
  	\subfigure[C]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_C.eps}
  	}
	\hspace{-5.00mm}
	\subfigure[Fe]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_Fe.eps}
	}
	\hspace{-5.00mm}
	\subfigure[FeS]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_FeS.eps}
  	}
	\hspace{-5.00mm}
	\subfigure[Mg$_2$SiO$_4$]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_Mg2SiO4.eps}
  	}
	\hspace{-5.00mm}
	\subfigure[MgO]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_MgO.eps}
  	}
	\hspace{-5.00mm}
	\subfigure[MgSiO$_3$]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_MgSiO3.eps}
  	}
	\hspace{-5.00mm}
	\subfigure[Si]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_Si.eps}
  	}
	\hspace{-5.00mm}
	\subfigure[SiO$_2$]{
  	\includegraphics[width=0.325\textwidth]{dustmassevolution_chi1000_SiO2.eps}
  	}
  \caption{Dust mass evolution (surviving mass) for the nine dust species tracked in this work versus time in units of cloud-crushing time.  The line styles correspond to the four different shock velocities, $10^{3}$~km~s$^{-1}$ (solid), $3 \times 10^{3}$~km~s$^{-1}$ (dot-dashed), $5 \times 10^{3}$~km~s$^{-1}$ (dashed), and $10^{4}$~km~s$^{-1}$ (dotted).   The colors correspond to the three different metallicities, $Z = 1~Z\subscript{\odot}$ (blue), $10~Z\subscript{\odot}$ (red), $100~Z\subscript{\odot}$ (green).  Panels (b), (c), and (h) do not fully span the range 0 to 1 on the $y$-axis.}\label{dustmassevol}
\end{figure*}

\begin{deluxetable}{lccc}
\tabletypesize{\scriptsize}
\tablecolumns{10}
\tablecaption{Final Dust-mass Survival Fraction \label{dustsurvival}}
\tablewidth{0pt}
\tablehead{\colhead{{\bf Simulation}} &
	\colhead{Al$_2$O$_3$} &
	\colhead{C} &
	\colhead{Mg$_2$SiO$_4$}}
\startdata
{\bf 1}	&	0.039	&	 0.947	&	0.879	\\
{\bf 2}	&	0.184	&	 0.953 	&	0.902	\\
{\bf 3}	&	0.661	&	 0.964	&	0.938	\\
{\bf 4}	&	0.001	&	 0.714	&	0.369	\\
{\bf 5}	&	0.001	&	 0.714 	&	0.371	\\
{\bf 6}	&	0.118	&	 0.659	&	0.409	\\
{\bf 7}	&	0.001	&	 0.610	&	0.181 	\\
{\bf 8}	&	0.001	&	 0.597 	&	0.171	\\
{\bf 9}	&	0.000 	& 	 0.531 	&	0.137 	\\
{\bf 10}	&	0.002	&	 0.585	&	0.120	\\
{\bf 11}	&	0.002	&	 0.568	&	0.113	\\
{\bf 12}	&	0.001	&	 0.442	&	0.067	\\
\vspace{-2.25mm}
\\
\tableline
\vspace{-2mm}
\\
{\bf \ Simulation}	&	MgSiO$_3$	&	SiO$_2$	&	MgO		\vspace{0.75mm}
\\
\tableline
\vspace{-1.5mm}
\\
{\bf 1}	&	0.413	&	 0.910	&	0.827	\\
{\bf 2}	&	0.518	&	 0.931	&	0.857	\\
{\bf 3}	&	0.789	&	 0.951	&	0.922	\\
{\bf 4}	&	0.038	&	 0.411	&	0.347	\\
{\bf 5}	&	0.039	&	 0.413	&	0.351	\\
{\bf 6}	&	0.162	&	 0.441	&	0.399	\\
{\bf 7}	&	0.014 	&	 0.203	&	0.184	\\
{\bf 8}	&	0.014	&	 0.192	&	0.177	\\
{\bf 9}	&	0.007	&	 0.156	&	0.152 	\\
{\bf 10}	&	0.018	&	 0.128	&	0.132	\\
{\bf 11}	&	0.016	&	 0.121	&	0.126	\\
{\bf 12}	&	0.008	&	 0.072	&	0.084	\\
\vspace{-2.25mm}
\\
\tableline
\vspace{-2mm}
\\
{\bf \ Simulation}	&	 Si 	&	 FeS 		&	 Fe 	\vspace{0.75mm}
\\
\tableline
\vspace{-1.5mm}
\\
{\bf 1}	&	0.998	&	0.643	&	 1.000	\\
{\bf 2}	&	0.998	&	0.708	&	 1.000	\\
{\bf 3}	&	0.993	&	0.816 	&	 0.989	\\
{\bf 4}	&	0.964	&	0.165	&	 0.959	\\
{\bf 5}	&	0.960	&	0.145	&	 0.940	\\
{\bf 6}	&	0.815	&	0.173 	&	 0.693	\\
{\bf 7}	&	0.849	&	0.055	&	 0.801	\\
{\bf 8}	&	0.822 	&	0.044 	&	 0.752 	\\
{\bf 9}	&	0.660	&	0.007	&	 0.465	\\
{\bf 10}	&	0.726 	&	0.031	&	 0.603	\\
{\bf 11}	&	0.688	&	0.025	&	 0.547	\\
{\bf 12}	&	0.446	&	0.005	&	 0.235
\enddata
\tablecomments{For simulation parameters, refer to Table \ref{sim_table}}
\end{deluxetable}

In studying the final (surviving) dust masses for each grain species, we find that the values are widely scattered for the simulations with the slower shock velocities.  The higher velocity simulations show dust survival of less than 20\% for six of the nine species, with C, Fe, and Si grains being the three species with the least destruction.  Although Fe grains had a higher survival rate than other dust species, the amount of Fe dust destruction was considerably higher than any of the simulations in our previous work.  In the most extreme case, only 24\% of the Fe dust mass survived.  If we look at dust survival for two other commonly studied grain species for the same extreme case, 44\% and 7\% of the initial dust mass remains for carbonaceous (C) and silicate (SiO$\subscript{2}$) grains, respectively.

For example, consider the differences in dust survival across grain species.  The difference, $\Delta \subscript{survival}$, between most destroyed grains and least destroyed grains ranges from as high as 96\% in Simulations 4 and 5 (Al$\subscript{2}$O$\subscript{3}$ compared to Si) to as low as 45\% in Simulation 12 (again, Al$\subscript{2}$O$\subscript{3}$ compared to Si).  This variation stems primarily from the initial distributions in grain radius.  A high degree of destruction is observed in the species with a majority of their mass locked up in small grains.  Destruction is reduced significantly for those species with appreciable mass in large grains.  As noted in Paper 1, grains with initial radii less than 0.1~$\mu$m are easily obliterated, while larger grains take a considerable duration to be substantially sputtered.  If the actual size distributions of dust grains in SNRs were greatly disparate from the distributions assumed here, the final dust mass survival fractions might change considerably.

\subsection{Density-Temperature Phase-Space}
In an attempt to understand the differences in the physical environments between the twelve simulations, we present phase-space diagrams of density and temperature to illustrate where the dust particles spend their time (Figure \ref{phaseplots}).  We also over-plot erosion rate contours for C grains as a function of density and temperature to determine where in phase-space the majority of the dust destruction occurs.  The most apparent difference between simulations is the rise in gas temperatures with increasing shock velocity, a direct result of shock-heating.  The increase in temperature pushes the gas into a regime of higher grain erosion, especially when the gas is still compressed to relatively high densities.  We also note that the large number of time steps at high temperature and low-density seen across all simulations is a result of the simulation end-state, at which point the cloud has been shock-heated and shredded.

\begin{figure*}[htp]
\centering
	\includegraphics[width=0.9\textwidth]{PhasePlots_MultiPanel_12_labeled_corrected.pdf}
	\caption{Phase plots of hydrogen number density ($n\subscript{H}$ in cm$^{-3}$) vs. temperature ($T$ in K).  The color scale represents the total number of times steps (N) spent by all dust particles in the simulation.  The contour lines show the dust destruction rate, log($da/dt$), in units of $\mu$m~yr$^{-1}$ for C grains. Each panel is one of the 12 simulations, labeled by the number in brackets [Table \ref{sim_table}].}\label{phaseplots}
\end{figure*}

In reviewing some of the subtle differences between simulations, we identify features that help to explain the differences between final dust masses.  First, we believe the high number of time steps spent by particles at high densities but very low temperatures in Simulation 3 ($v\subscript{shock} = 10^{3}$~km~s$^{-1}$, $Z = 100~Z\subscript{\odot}$) is likely the dominating factor in the decreased dust destruction when compared to the lower metallicity simulations (1 and 2). In this scenario, the temperatures of the gas drop low enough from the radiative cooling to slow grain erosion.  Although Simulation 6 ($v\subscript{shock} = 3 \times 10^{3}$~km~s$^{-1}$, $Z = 100~Z\subscript{\odot}$) shows a similar population of high-density, low-temperature dust particle environments, it is not sufficient to produce the same consistent decrease in dust destruction, owing primarily to the overall increase in gas temperature.  The pile up of dust particles at $T = 1000$~K, seen in Simulations 3 and 6, is a direct result of the temperature floor described in Section \ref{modifications}.  For $v\subscript{shock} = 5 \times 10^{3}$~km~s$^{-1}$, there is a distinct feature present in Simulation 9 with $Z = 100~Z\subscript{\odot}$ that separates it from the lower metallicity cases.  As a result of the enhanced metal cooling, the gas is able to cool and condense beyond a hydrogen number density of $n\subscript{H} = 10^{4}$ cm$^{-3}$, but it remains within a temperature regime to allow for appreciable grain erosion, explaining the increased dust destruction observed in the dust mass evolution.

In the case of the highest velocity shock, the intense heating appears to overwhelm the effects of radiative cooling, even for the highest metallicity simulations, as the phase plots for Simulations 10-12 appear nearly identical in the qualitative sense.  Given the similarity between the physical environments experienced by the dust particles, it would appear that in this regime the dominant cause for increased dust destruction at high metallicity is the increase in the erosion rates.  This is not an unreasonable explanation, since the dust particles spend the majority of their time above 10$^{6}$~K, the point at which the dust destruction rates begin to vary significantly for higher metal abundances.

\section{Summary and Discussion}\label{summary}
The above results can be summarized as follows:
\begin{itemize}
	\item{The degree of destruction varies widely across the explored parameter space. For the least destructive cases (low shock velocity, high gas metallicity), we see near-negligible grain destruction for some species (Fe and Si) and the highest survival rates for all other species. In the maximally destructive cases (high shock velocity, high gas metallicity), we find complete to near-complete destruction for multiple grain species (Al$_2$O$_3$, FeS, and MgSiO$_3$).}
	\item{The relative velocity between the reverse shock and the cloud of ejecta material is a dominant factor in determining the survival rate of dust grains; an increase in velocity correlates directly with an increase in dust destruction.  The most extreme example is for silicate (SiO$\subscript{2}$) grains, where an additional 88\% of the initial dust mass is destroyed between the simulations with $v\subscript{shock} =  10^{3}$~km~s$^{-1}$ and those with $v\subscript{shock} =  10^{4}$~km~s$^{-1}$.}
	\item{There appears to be a threshold shock velocity between 1000 and 3000 km s$^{-1}$ at which the amount of dust destruction at early times ($t < t\subscript{cc}$) is increased considerably.}
	\item{In the physically expected, highly metal-enriched simulations, we find ranges of dust mass survival fraction of 44-96\%, 7-95\%, and 24-99\% for the often studied grain species of C, SiO$\subscript{2}$, and Fe, respectively.}
	\item{As in Paper I, the initial distribution in grain radius greatly influences the overall survival of a given species, with grains with initial radii below 0.1~$\mu$m being destroyed in most simulations.}
	\item{For the high-metallicity ($Z = 100~Z\subscript{\odot}$) simulations, the balance between the increased radiative cooling and increased sputtering yield depends strongly on the speed of the shock.  When the shock velocity is below 3000~km~s$^{-1}$, the increased cooling lowers the temperature of the gas enough to suppress dust destruction.  At high velocities, the gas temperature is driven up to a high-erosion regime.}
\end{itemize}

In comparing this new suite of simulations to that of Paper I, the key differences are the higher metallicity cooling functions, the erosion rates for enhanced metal abundances for all elements up to iron, the exploration of higher shock velocities ($v\subscript{shock} = 10^{4}$~km~s$^{-1}$), and the suppressed cooling in the ambient medium surrounding the cloud.  As a consistency check, we find that the simulations with $Z = Z\subscript{\odot}$ for shock velocities of $v\subscript{shock} = 1000$~km~s$^{-1}$, 3000~km~$^{-1}$, and 5000~km~s$^{-1}$ produce very similar dust mass survival fractions, indicating that these changes do not invalidate our previous results.  In fact, the degree of dust survival does not begin to deviate significantly from Paper 1 until the metallicity is increased to $Z = 100~Z\subscript{\odot}$, as expected.

We also find that our dust survival fractions remain in agreement with observations of IR emission in SNRs.  Specifically, the continued survival of Si grains matches the high abundances found in Cas A by \cite{Rho:2008qf}, while complete destruction of Al$_2$O$_3$ grains and survival of amorphous carbon grains match observations of SNR 1E0102-7219 by \cite{Sandstrom:2009fk}.

Ideally, we would have liked to explore a metallicity regime in which the gas in the ejecta cloud is saturated with metals, effectively mimicking clouds of pure oxygen, silicon, sulfur, argon, or calcium.  Using oxygen as an example, an abundance ratio of order one would require a metallicity of $Z \sim 1000~Z\subscript{\odot}$ since the solar abundance ratio is $n\subscript{O}/n\subscript{H} \sim 5\times 10^{-4}$.  Unfortunately, due to the extremely short cooling times, this proved computationally prohibitive.  The simplest solution is to limit the simulation time-step by the cooling time and allow the simulation to run for a longer physical time, but it is also the most computationally expensive.  In order to probe the highest metallicity regimes, a more resource-efficient method would need to be implemented without sacrificing the validity of the simulation.  Such a solution is left to future work.

Currently, we still operate in the limit of strictly thermal sputtering, since our dust grains are directly coupled to the flow of the gas.  Because large grains would require a greater transfer of momentum from the gas before becoming completely entrained in the flow, it is possible that the dominant mode of sputtering for such grains would be in the non-thermal limit, as high velocity gas washes over them.  In order to decouple the dust particles from the motions of the gas, we would have to track separated grain populations of varying radii.  The number of radius bins required to accurately track the entire dust mass for all nine grain species is uncertain, but it could be computationally significant.  As mentioned in Paper I, it may not be unreasonable for the dust particles to be coupled to the flow if the grains are charged and magnetic fields permeate the ejecta material.  However, magnetized ejecta clouds would require an investigation of the effects of betatron acceleration on charged grains \citep{Shull:1977lr, Shull:1978fk}.

Finally, we investigated the possibility of transitioning from post-processing of our dust particle histories to compute erosion to an ``on-the-fly" method that would sputter the grains as the simulation ran.  This would allow metals that are sputtered off the grains to feed back into the ambient medium and increase the metallicity.  However, if we assume that the gas within the ejecta cloud is already heavily metal-enriched, the amount of metals released into the gas in this way would not alter the metallicity enough to make a substantial difference.  As we have shown, it takes a large change in the metal abundance to create an significant difference in the amount of dust that survives the shock-cloud interaction.

\acknowledgements
We thank the anonymous referee for comments that helped to clarify and strengthen this work.  We also thank Robert Fesen for useful discussion and comments.  The computing resources needed for this work were provided by NASA HEC Allocation SMD-10-2612 on the supercomputer, {\it Pleiades}, at the NASA Advanced Supercomputing Facility.  This project was funded by NSF Theory Grant AST07-07474, STScI Archive Theory Grant AR 11774.01-A, and the NSF Graduate Research Fellowship Program (D.W.S.).

redpajama_set_name: RedPajamaArXiv