text:
\section{Introduction}
Blue Straggler Stars (BSSs) represent the most numerous exotic population (i.e. not explainable
in terms of passive evolution of single stars) in stellar systems. They are observed in globular (GCs; see for example
\citealt{ferraro03,ferraro12,ferraro18,dalessandro08a}) and open clusters \citep{mathieu09} as well as in dwarf galaxies \citep{monelli12}.
BSSs appear hotter and brighter than Turn-Off (TO) stars in optical colour-magnitude diagrams (CMDs)
of stellar systems \citep{ferraro92,ferraro93,ferraro97},
thus mimicking a sparse sequence of younger and more massive objects. Observational evidence (see for example
\citealt{shara97,ferraro06a,fiorentino14,brogaard18}) showed that indeed BSSs are up to twice more massive than TO stars in their
host clusters.

BSSs can either be the result of mass accretion
between two stars in a binary system \citep{mccrea64,zinn76} or the end products of stellar mergers induced
by collisions between single stars or binary systems \citep{hills76}.
While the first process is common to any stellar environment being the result of the long-term evolution of binary systems, the second requires high density environments. As a consequence, in GCs, where the stellar density varies significantly from the center to the external regions, BSSs
can be generated by both processes with relative efficiencies depending on the local
density \citep{fusipecci92,ferraro99}. In the high crowded cores of stellar clusters, both formation channels are expected
to be active, although the mass-transfer process seems to be the most efficient one (see for example \citealt{knigge09}).

Interestingly, significant support to the co-existence of both collisional and mass-transfer BSSs in the same system,
comes also from the detection of a double BSS sequence separated in color and magnitude
in four post-core collapse Galactic GCs, namely M~30 \citep{ferraro09}, NGC~362 \citep{dalessandro13}
and M~15 (G. Beccari et al. 2018, in preparation), and NGC~1261,
which has been suggested to be in a post-core collapse bounce state \citep{simunovic14}.
The fact that the two sequences are well separated in the CMDs suggests that they were generated by a recent and short-lived event instead
of a continuous formation process.

\citet{ferraro09} proposed that the origin of the double sequence can be related to
the core-collapse (CC) process that is expected to largely enhance the probability of
collisions over a relatively short period of time (of the order of few tens of Myr), thus promoting the formation of collisional BSSs.
In fact, during CC the central density rapidly increases, thus producing an increase in gravitational interactions
\citep{meylan97} able to trigger the formation of new BSSs through both direct stellar collisions and mass
transfer activity induced by dynamical encounters involving primordial binaries
(e.g., \citealt{leonard89,hurley05,banerjee16}).
Thus, the CC process could in principle generate a nearly coeval population of BSSs with different masses
that in the CMD appears as a  well-defined, tight sequence in addition to the ``normal'' (spread)
BSS population constantly generated by the mass transfer process in primordial binaries,
which are present in any environment.
Indeed, the location of the collisional sequences in the CMD can be well reproduced
by collisional models \citep{sills09}. Analogously to what is commonly done in the Main Sequence
fitting dating method, this allows us to use the extension in luminosity of the collisional sequence
to date the epoch of the formation of the collisional BSSs, thus providing an hint to the epoch of
CC in the parent cluster (\citealt{ferraro09}).
Following these arguments \citet{ferraro09} and \citet{dalessandro13} concluded that the CC event
occurred recently (1-2 Gyr ago) in both M~30 and NGC~362.
The red BSS sequence is instead compatible with
models of binary systems undergoing mass-transfer (\citealt{tian06,xin15}, but see the discussion in \citealt{jiang+17}).

Recently, \citet{li18} have claimed that also the young ($t\sim2$ Gyr)
GC NGC~2173 in the Large Magellanic Cloud (LMC) shows a double sequence of BSSs.
NGC~2173 is the only non post-core-collapse cluster suggested to show a double BSS sequence so far. This result can have therefore important implications
in our understanding of BSS double sequence formation.
In fact, because of the cluster's low central density and age, the authors argued that collisions are not a
viable scenario to explain the blue BSS sequence in this case. On the contrary, they suggest
that most likely the two sequences are populated by binaries with different mass-ratios. However,
they admit that it is unclear how this scenario can reproduce the observed discrete color
and magnitude distribution.

The significance of the observational results presented by \citet{li18} is strongly hampered by the fact
that they do not take into account field star contamination.
In LMC clusters, field interlopers can potentially have a large impact on the BSS region in the CMD, because of the
large young LMC stellar component.
In this work we re-analyse the BSS population properties and distribution taking into account the effect of field star
contamination. To this aim, we used two different HST data-sets, sampling both the cluster central regions
and the surrounding LMC field. We find that about half of the BSS total sample used by \citet{li18}
is actually populated by likely non-member stars.
More interestingly, we show that the decontamination analysis essentially remove all the BSS along the
red sequence, thus demonstrating that the double sequence claimed by \citet{li18} is essentially an
artefact due to the contamination of the LMC field stars.

\begin{figure}
\includegraphics[width=9cm]{f1.pdf}
\caption{Map of the WFC3/UVIS and ACS/WFC fields of view. The cluster core, half-light
and tidal radii (see Appendix) are shown in red. }
\label{fig:map}
\end{figure}

\section{Observations and data analysis}
We used a combination of Hubble Space Telescope Wide Field Camera 3 - UVIS channel (WFC3/UVIS) and
Advanced Camera for Survey - WFC (ACS/WFC) images obtained through proposal GO12257 (PI: Girardi).
WFC3/UVIS data represent the primary observations and sample the cluster innermost regions
({\it Cluster Pointing}), while ACS/WFC data have been
obtained as parallel observations ({\it Field Pointing}) and sample a region located at $\sim300\arcsec$ southward
from the cluster (see Figure~\ref{fig:map}).

\begin{table*}
\begin{center}
\caption{Summary of the HST data-set used in this work.}
\begin{tabular}{ c  c  c  c }
\hline
\hline
Instrument & Filter & ${\rm t}_{{\rm exp}}$ (s) & Camera\\
\hline
{\small Cluster Pointing}      & {\scriptsize F336W} &  { \scriptsize $1 \times 800 + 2\times 700$}    & { {\small WFC3/UVIS}}\\
{}			       & {\scriptsize F475W} &  { \scriptsize $1 \times 120 + 2\times 700$}    & {}\\
{}			       & {\scriptsize F814W} &  { \scriptsize $1 \times 20 +  3\times 700$}    & {}\\

\hline
{ \small Field Pointing}	  & { \scriptsize F475W}  & { \scriptsize $2 \times 500 + 1 \times 700$}  & {{\small ACS/WFC}}\\
{}			    & { \scriptsize F814W}  & { \scriptsize $1 \times 20 + 1 \times 600 + 1\times 690+ 2\times 700$}	  & {}\\
\hline

\hline
\hline
\end{tabular}
\\
\label{dataset}
\end{center}
\end{table*}

The most relevant details about the adopted sets of images are summarised in Table~1.
For both data-sets, an appropriate dither pattern of a few arcseconds has been adopted for each pointing
in order to fill the inter-chip gaps and avoid spurious effects due to bad pixels.

\begin{figure}
\includegraphics[width=9cm]{f2.pdf}
\caption{($m_{F475W},m_{F475W}-m_{F814W})$ CMDs of the {\it Cluster Pointing} and {\it Field Pointing}.
The black boxes highlight the BSS region that has been used in the analysis (see Section~3). }
\label{fig:opt-cmds}
\end{figure}

The photometric analysis was performed on the {\tt FLC} images (which are corrected for charge transfer efficiency)
{\tt pixel area
map} corrected images by using {\tt DAOPHOT IV} \citep{stetson87} and relative routines, following
the approach used in other works of our group
(see \citealt{dalessandro18a,dalessandro18b} for a recent reference).
Briefly, point spread function (PSF) models were derived for each image and chip by using some tens of stars and they were applied to stars whose flux peaks are above $3\sigma$ from the local background.
A master list including stars detected in at least four images was then created.
At the corresponding positions of stars in the master-list, a fit was forced with
{\tt DAOPHOT/ALLFRAME} \citep{stetson94} in each frame of the two data-sets.
For each star thus recovered, multiple magnitude estimates obtained in each chip were homogenised by using
{\tt DAOMATCH} and {\tt DAOMASTER}, and their weighted mean and standard deviation were finally adopted as
star magnitude and photometric error.

Instrumental magnitudes of both the WFC3 and ACS catalogs
were calibrated onto the VEGAMAG photometric system by using the recipes and zero-points reported in the HST web-sites.
Instrumental coordinates were first corrected
for geometric distortions by using equations
by \citet{bellini09} for the WFC3/UVIS data and the most updates Distortion Correction Tables
({\tt IDCTAB}) provided on the dedicated page of the Space Telescope Science Institute for the ACS/WFC images.
Then they were reported to the absolute coordinate system ($\alpha, \delta$) as defined by the World Coordinate System
of the HST images.
The final color magnitude diagrams for both data-sets are shown in Figure~\ref{fig:opt-cmds}.
We note that, in agreement with \citet{li18}, a double BSS sequence can be easily distinguished in the {\it Cluster Pointing} CMD.

\section{Density profile and structural parameters}
In order to derive cluster structural parameters
we built the cluster number density profile by using both the {\it Cluster} and the {\it Field} HST data-sets.
These information, and the extension of the cluster in particular, are key for the decontamination procedure described in
Section~4.

The analysis has been performed following the procedure fully described in Miocchi et al. (2013; see also \citealt{lanzoni10}). We used as center of the cluster the one obtained by \citet{li18}.
We considered 11 concentric annuli centred on the cluster center, each one divided into a number of sub-sectors ranging from two to four.
In each sub-sector,
we counted the number of stars with $m_{F606W}<21$.
The projected stellar density in each annulus
is the mean of the values measured in each sub-sector and the uncertainty has been estimated from the variance
among the sub-sectors.
The observed density profile is shown in Figure~\ref{fig:dens} (open circles). It smoothly declines as a function of distance
and then it flattens at distance $d>200\arcsec$ due to the contribution
of the LMC field background. This has been estimated by averaging the two outermost values,
and it has been subtracted to the observed distribution to obtain the cluster (decontaminated) density
profile (filled circles).

We then derived the cluster structural parameters
by fitting the observed density profile with a spherical, isotropic, single-mass \citet{king66} model. The best-fit model
results in a cluster with a King dimensionless potential $W_0 = 6.1$, corresponding to a concentration parameter $c=1.28$
a core radius $r_c = (11.1\pm1.2)\arcsec$, a half-mass radius $r_h=(33.2\pm3.3)\arcsec$,
a tidal radius $r_t=(235.6\pm23.5)\arcsec$. Assuming a distance to the LMC of 50 kpc \citep{pietr13}, these values correspond to 2.7pc, 8.1pc and 57.1pc respectively.
While the derived values of $r_c$ and $r_h$ are in quite good agreement with literature estimates, $r_t$ results to be significantly smaller than that obtained by \citet{vdm05} and \citet{li18}, who quote
95.5pc and 131.30pc, respectively.
Such a discrepancy is likely due to the lack of an appropriate control area allowing to properly
sample and subtract the LMC background.

\begin{figure}
\includegraphics[width=9cm]{2173_prof.pdf}
\caption{Observed star count density profile of NGC2173 (open circles).
The dashed line represents
the density value of the background as derived by averaging the two outermost radial bins. The black filled dots are densities obtained after background subtraction. The best-fit single-mass King model is overplotted to the observations (black solid line). The lower panel shows the residuals between the observations and the best-fit model.}
\label{fig:dens}
\end{figure}

\begin{figure}
\includegraphics[width=9cm]{f4.pdf}
\caption{($m_{F475W},m_{F475W}-m_{F814W})$ CMD of the {\it Field Pointing} (in red) superimposed to the
{\it Cluster Pointing} CMD (in black) for a direct comparison. Clearly, the SGB/RGB at $m_{F475W}-m_{F814W}>0.8$ and $m_{F475W}>21$,
as well as the BSS region are significantly contaminated by LMC stars.}
\label{fig:comp-cmds}
\end{figure}

\section{Color magnitude diagrams decontamination}
The CMDs of the vast majority of stellar clusters in the Magellanic Clouds are strongly contaminated
by field star interlopers. In fact, as shown in Figure~\ref{fig:opt-cmds}
for the case of NGC~2173, cluster evolutionary sequences are overlaid to similarly populated LMC
star sequences. As a consequence, a proper decontamination is key  when studies on specific portions
and features in clusters' CMDs are performed and/or when stellar population number counts are concerned.

Unfortunately, given the distance of the Magellanic Clouds a detailed separation between field and cluster stars based on
proper motion is possible only for a few cases. Moreover accurate Gaia DR2 proper motions
data are available only for the brightest stars.
As a consequence, to assess the impact of field contamination on the CMD of NGC~2173 we used a  statistical approach based
on the comparison between the distribution of stars in the cluster CMD, at different radial distances from the cluster center,
and that of a region representative of the surrounding LMC field.
As shown in Figure~\ref{fig:map}, the available ACS/WFC {\it Field Pointing} is ideal to perform a suitable decontamination
of the program cluster. In fact, the ACS/WFC FOV provides a relatively large area (which is $\sim50\%$ larger
than the WFC3/UVIS
{\it Cluster Pointing} FOV) to sample the LMC star distribution. Moreover, it is located beyond the cluster tidal
radius (see Section~3) thus ensuring that any contribution from cluster stars (if any) is negligible.

\begin{figure}
\includegraphics[width=9cm]{f5_new.pdf}
\caption{Comparison between the observed {\it Cluster Pointing} CMD and one resulting from the statistical decontamination procedure
described in Section~3.}
\label{fig:deco}
\end{figure}

\begin{figure*}
\centering
\includegraphics[width=14cm]{f6.pdf}
\caption{Nine decontaminated CMDs obtained as described in Section~3, using different grid's cell widths and limits.
In all cases there is no significant evidence of a double BSS sequence.}
\label{fig:multi}
\end{figure*}

A first clue about the level of the LMC field contamination affecting the {\it Cluster} CMD can be obtained by
applying the method described in \citet[][see also \citealt{ziri16} for a similar application]{knoetig14}
which allows us
to calculate the probability of a star in the {\it Cluster Pointing} CMD to belong to the field population.
We divided both the {\it Cluster} and the {\it Field} CMDs in a regular grid of $0.3\times0.25$~mag$^2$
and we counted the number of stars falling within each cell.
The number counts of likely field stars were then corrected for the different
area covered by the two data-sets. We then used equations 23 in \citet{knoetig14}
to estimate the probability that stars observed in a given cell are cluster members.
We refer the reader to that paper for all relevant details about the method.

This approach has the advantage to be applicable to cells with small and large numbers.
The analysis shows that there are at least two regions in the {\it Cluster} CMD significantly contaminated by foreground LMC stars:
the one including stars with ($m_{F475W}-m_{F814W})>0.8$ and $m_{F475W}>21$, where at least two LMC
sub-populations older than NGC~2173 can be distinguished, and the BSS region, in which
the member probability is below $20\%$.

In Figure~\ref{fig:opt-cmds} we highlight with a rectangular box the BSS region
in the ($m_{F475W},m_{F475W}-m_{F814W}$) CMDs of both the {\it Cluster} and the {\it Field} pointings.
Even from a preliminary visual inspection it is evident that a significant contribution of the LMC field
is expected in that region. In fact, we count 37 stars ($\pm6.2$ as Poissonian error) in the WFC3 {\it Cluster Pointing}
and 21 in the ACS/WFC {\it Field Pointing}. After rescaling this latter value to the WFC3/UVIS area,
we obtain a total number of contaminants of $15\pm3.9$, which yields a possible number of likely BSS cluster members
of $22\pm7.2$ in the {\it Cluster Pointing} (corresponding to roughly $60\%$ of the observed sample).

In order to allow a direct comparison between the cluster and the field sequences, in Figure~\ref{fig:comp-cmds},
we co-added the two CMDs, highlighting in red the stars measured in the {\it Field Pointing}.
The matching of the sequences is impressive
and clearly indicates the regions of the CMD which are mainly affected by the LMC field contamination,
fully confirming the analysis discussed above.
In fact, it is evident that the two SGB/RGB visible at  ($m_{F475W}-m_{F814W})>0.8$ and $m_{F475W}>21$
are essentially due to the LMC field. Moreover, both the Red Giant Branch and the Helium Red Clump of the cluster
are affected by field contamination, as well as the BSS region.
In particular, from the comparison it also emerges that the BSSs in the {\it Field} and the {\it Cluster} CMDs are
distributed in a different way.
In fact, field candidate BSSs (or young populations)
tend to be preferentially distributed in the brightest and
reddest portion of the box, thus suggesting that red BSSs in the {\it Cluster} CMD are less likely to be cluster
members than the blue ones.

These simple arguments demonstrate that a very detailed analysis is needed to
assess the possible presence of a double sequence, as that claimed by \citet{li18}.

To this aim we have statistically decontaminated the ($m_{F475W},m_{F475W}-m_{F814W}$) CMD by using the following approach.
We split the {\it Cluster Pointing} in five concentric annuli centred on the cluster center.
We then partitioned the CMD corresponding to each annulus in the same grid of cells discussed above for the {\it Field Pointing} CMD.
The same number of stars counted in the corresponding
cell of the {\it Field} CMD was then randomly removed from the {\it Cluster} cell,
accounting for the difference in the size between the area covered by ACS and the WFC3 pointings.
The stars surviving this analysis correspond to the likely cluster members and constitute the decontaminated CMD.
The original CMD and the one resulting from the decontamination analysis are shown in Figure~\ref{fig:deco}
(to ease the comparison with \citealt{li18} we show the results in the ($m_{F336W},m_{F336W}-m_{F814W}$) CMD).
The analysis has efficiently removed highly probable LMC stars and thus the evolutionary sequences appear to be much
better defined in the decontaminated CMD.
As expected, from the arguments discussed above,
only about 25 BSSs survived the decontamination analysis.
More importantly, the remaining BSS population is quite scarce and shows no evidence of a double sequence.

We performed the above analysis several tens of times by changing the cells' dimensions and the grids' limits.
Some of the resulting decontaminated CMDs are shown in Figure~\ref{fig:multi} as examples.
As expected, while the exact number and position of the surviving BSSs may differ from
one realisation to the other, the two main results remain:
i) the total fraction of BSS cluster members in the {\it Cluster Field} is only about $60\%$ of the observed one;
ii) in none of the realisations we find evidence of a double BSS sequence.

\section{Summary and Conclusions}
We used observations available in the HST Archive
to perform a detailed decontamination of the CMD of the young cluster NGC~2173 in the LMC, with the specific aim of
assessing the existence of the double BSS sequence as recently claimed by \citet{li18}. Our analysis demonstrates
that the detected feature is an artefact due to the contamination by LMC field stars.
NGC~2173 turns out to be a young cluster with a single and poorly populated BSS sequence, which can likely be the result of binary evolution.
As a consequence, the case of NGC~2173 is not relevant for the understanding and the discussion on the origin of the double
BSS sequences observed so far in a few old globular clusters (\citealt{ferraro09,dalessandro13,simunovic14}, Beccari et al. 2018).

We used the decontaminated CMD shown in the right panel of Figure~\ref{fig:deco} to select
likely member BSSs and reference populations.
We find that the BSS radial distribution in NGC~2173 is consistent (within the uncertainties related to the
small number of stars and the decontamination process) with that of normal Turn Off cluster stars.
This is not surprising since the BSS originated by the evolution of primordial binaries are expected to have,
at their origin, a radial distribution indistinguishable from other lighter stars.
Such a radial distribution can be maintained for a long time in dynamically unevolved clusters,
where the time-scale of the BSS sedimentation toward the cluster center due to dynamical friction is particularly long.
This is indeed observed in many dynamically young Galactic clusters (see the cases of Omega Centauri \citealt{ferraro06b},
NGC~2419 and NGC~6101 \citealt{dalessandro08b,dalessandro15}) even after several Gyr from their formation
(see the discussion in \citealt{ferraro12,ferraro18,lanzoni16}). Hence
no peculiar phenomena need to be advocated to explain the BSS population of NGC~2173.
Instead this case clearly points out the importance and the necessity of an appropriate field
decontamination to properly assess any feature or star population counts in heavily contaminated stellar
aggregates, as the stellar clusters in the LMC.

\vspace{1.0cm}
ED acknowledges financial support from the Leverhulme Trust Visiting Professorship Programme VP2-2017-030.
ED is also grateful for the warm hospitality of LJMU where part of this work was performed.
NB gratefully acknowledges funding from the ERC under the European Union's Horizon 2020 research and innovation programme via
the ERC Consolidator Grant Multi-Pop (grant agreement number 646928, PI Bastian). NB is a Royal Society University Research Fellow.
The authors thank the anonymous referee for his careful reading of the paper and useful comments.

redpajama_set_name: RedPajamaArXiv


text:
Home / News & Sport / Storm Darwin passes, as 190,000 without power
Storm Darwin passes, as 190,000 without power
More than 2,000 ESB engineers are working to restore power to 190,000 homes and businesses who are without power around the country.
The crews are being assisted by staff from Northern Ireland Electricity.
However ESB is warning it will be late into this evening before supply is restored to most areas and a large number of customers will still be without power tonight.
Mike Fitzgerald of ESB Networks says the areas worst affected are in the south and west where parts of the network will have to be completely rebuilt.
"We have all resources at this stage in the field which involves approximately 2,000 of our own staff, supplemented with our colleagues from NIE and from contractors" he said.
"So all resources are being deployed; staff from less-affected areas of the country are moving down to those worst-affected areas".
"So everything is being done to connect our customers as soon as possible" he added.

redpajama_set_name: RedPajamaCommonCrawl


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text:
\section{Introduction}

In a series of papers starting with \cite{andrewspa1}, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's $\Omega$ operator \cite{macmahon} to systematically compute generating functions related to various families of integer partitions.
Andrews et al's papers concern generating functions of the form
\[
  f_P (z_1, \dots, z_n) := \sum_{ \lambda \in P } z_1^{ \lambda_1 } \cdots z_n^{ \lambda_n }
  \qquad \text{ and } \qquad
  f_P (q) := f_P (q, \dots, q) = \sum_{ \lambda \in P } q^{ \lambda_1 + \dots + \lambda_n } ,
\]
for some set $P$ of \emph{partitions} $\lambda = (\lambda_1, \dots, \lambda_n)$; i.e., we think of the integers $\lambda_n \ge \dots \ge \lambda_1 \ge 0$ as the \emph{parts} when some integer $k$ is written as $k = \lambda_1 + \dots + \lambda_n$.
If we do not force an order onto the $\lambda_j$'s, we call $\lambda$ a \emph{composition} of $k$.
Below is a sample of some of these striking results.

\begin{theorem}[Andrews \cite{andrewsPA2}] \label{higherdiffthm}
Let
\[
  P_r := \left\{ \lambda : \, \sum_{ j=0 }^t (-1)^j \binom t j \lambda_{ k+j } \ge 0 \ \text{ \rm for } \ k \ge 1, \ 1 \le t \le r \right\}
\]
(where we set undefined $\lambda_j$'s zero).
Then
\[
  f_{ P_r } (q) = \prod_{ j=1 }^\infty \frac{ 1 }{ 1 - q^{ \binom{ j+r-1 }{ r } } } \, .
\]
In words: the number of partitions of an integer $k$ satisfying the ``higher-order difference conditions" in $P_r$ equals the number of partitions of $k$ into parts that are $r$'th-order binomial coefficients.
\end{theorem}

\begin{theorem}[Andrews--Paule--Riese \cite{andrewspauleriesePA9}] \label{ngonpartitionthm}
Let $n \ge 3$ and
\[
  \tau := \left\{ (\lambda_1, \dots , \lambda_n) \in \mathbb{Z}^n : \, \lambda_n \ge \cdots \ge \lambda_1 \ge 1 \ \text{ \rm and } \ \lambda_1 + \cdots + \lambda_{n-1} > \lambda_n \right\} ,
\]
the set of all ``$n$-gon partitions."
Then
\[
  f_{ \tau } (q) =  \frac {q^n}{(1-q)(1-q^2) \cdots (1-q^n)} - \frac{ q^{2n-2} }{ (1-q) (1- q^2) (1 - q^4) (1 - q^6) \cdots (1 - q^{2n-2}) } \, .
\]
More generally,
\begin{align*}
  f_{ \tau } (z_1, \dots, z_n)
  &=  \frac {Z_1}{(1-Z_1)(1-Z_2) \cdots (1-Z_n)}\\
  &\qquad - \frac{ Z_1 Z_n^{n-2} }{ (1-Z_n) (1- Z_{n-1}) (1 - Z_{n-2} Z_n) (1 - Z_{n-3} Z_n^2) \cdots (1 - Z_1 Z_n^{n-2}) }
 \, ,
\end{align*}
where $Z_j := z_j z_{ j+1 } \cdots z_n \text { for } 1 \leq j \leq n$.
\end{theorem}

The composition analogue of Theorem \ref{ngonpartitionthm} was inspired by a problem of Hermite \cite[Ex.~31]{polyaszego}, which is essentially the case $n=3$ of the following.

\begin{theorem}[Andrews--Paule--Riese \cite{andrewspauleriesePA3}] \label{hermitethm}
Let
\[
  H := \left\{ (\lambda_1, \dots , \lambda_n) \in \mathbb{Z}_{ >0 }^n : \, \lambda_1 + \dots + \widehat{ \lambda_j } + \dots + \lambda_n \ge \lambda_j \ \text{ \rm for all } \ 1 \le j \le n \right\} \, .
\]
Then
\[
  f_H (q) = \frac{ q^n }{ (1-q)^n } - n \, \frac{ q^{ 2n-1 } }{ (1-q)^n (1+q)^{ n-1 } } \, .
\]
\end{theorem}

A natural question is whether there exist ``full generating function" versions of Theorems \ref{higherdiffthm} and \ref{hermitethm}, in analogy with Theorem \ref{ngonpartitionthm}; we will show that such versions (Theorems \ref{higherdiffthmgeneralized} and \ref{fullhermitethm} below) follow effortlessly from our approach. (Xin \cite[Example 6.1]{xinmacmahon} previously computed a full-generating-function related to Theorem \ref{hermitethm}.)

Our main goal is to prove these theorems geometrically, and more, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.
This approach is not new:
Pak illustrated in \cite{pakpartitionprocams,pakpartitionsurvey} how one can obtain bijective proofs by realizing when both sides of a partition identity are generating functions of lattice points in unimodular cones (which we will define below); this included most of the identities appearing in \cite{andrewsPA2}, including Theorem~\ref{higherdiffthm}.
Corteel, Savage, and Wilf \cite{corteelsavagewilf} implicitly used the extreme-ray description of a cone (see Lemma \ref{linalglemma} below) to derive product formulas for partition generating functions, including those appearing in \cite{andrewsPA2}.
Beck, Gessel, Lee, and Savage \cite{beckgesselleesavage} used triangulations of cones to extend results of Andrews, Paule, and Riese \cite{andrewspauleriesePA7} on ``symmetrically constrained compositions.''
However, we feel that each of these papers only scratched the surface of a polyhedral approach to partition identities, and we see the current paper as a further step towards a systematic study of this approach.

While the $\Omega$-operator approach to partition identities is elegant and powerful (not to mention useful in the search for such identities), we see several reasons for pursuing a geometric interpretation of these results.
As discussed in \cite{corteelleesavage5guidelines}, partition analysis and the $\Omega$ operator are useful tools for studying partitions and compositions defined by linear constraints, which is equivalent to studying integer points in polyhedra.
An explicit geometric approach to these problems often reveals interesting connections to geometric combinatorics, such as the connections and conjectures discussed in Sections~\ref{cayleycomp} and~\ref{hypersimplices} below.
Also, one of the great appeals of partition analysis is that it is automatic; Andrews discusses this in the context of applying the $\Omega$ operator to the 4-dimensional case of lecture-hall partitions in \cite{andrewspa1}:
\begin{quote}
The point to stress here is that we have carried off the case $j=4$ with no effective combinatorial argument or knowledge.
In other words, the entire problem is reduced by Partition Analysis to the factorization of an explicit polynomial.
\end{quote}
As we hope to show, the geometric perspective can often provide a clear view of sometimes mysterious formulas that arise from the symbolic manipulation of the $\Omega$ operator.

\section{Polyhedral cones and their lattice points}\label{latticepointsection}

We use the standard abbreviation ${\mathbf z}^{\mathbf m} := z_1^{ m_1 } \cdots z_n^{ m_n }$ for two vectors ${\mathbf z}$ and ${\mathbf m}$.
Given a subset $K$ of $\mathbb{R}^n$, the \emph{(integer-point) generating function} of $K$ is
\[
  \sigma_K (z_1, \dots, z_n) := \sum_{ {\mathbf m} \in K \cap \mathbb{Z}^n } {\mathbf z}^{\mathbf m} \, .
\]
We will often encounter subsets that are cones, where a \emph{(polyhedral) cone} $C$ is the intersection of finitely many (open or closed) halfspaces whose bounding hyperplanes contain the origin.
(Thus the cones appearing in this paper will not all be closed but in general partially open.)
A closed cone has the alternative description (and this equivalence is nontrivial \cite{ziegler}) as the nonnegative span of a finite set of vectors in $\mathbb{R}^n$, the \emph{generators} of~$C$.

An $n$-dimensional cone in $\mathbb{R}^n$ is \emph{simplicial} if we only need $n$ halfspaces to describe it.
All of our cones will be \emph{pointed}, i.e., they do not contain lines.
The following exercise in linear algebra shows how to switch between the generator and halfspace descriptions of a simplicial cone.

\begin{lemma}\label{linalglemma}
Let ${\mathbf A}$ be the inverse matrix of ${\mathbf B} \in \mathbb{R}^{ n \times n }$. Then
\[
  \left\{ {\mathbf x} \in \mathbb{R}^n : \, {\mathbf A} \, {\mathbf x} \ge {\mathbf 0} \right\} = \left\{ {\mathbf B} \, {\mathbf y} : \, {\mathbf y} \ge {\mathbf 0} \right\} ,
\]
where each inequality is understood componentwise.
\end{lemma}

The (integer-point) generating function of a simplicial cone $C \subset \mathbb{R}^n$ can be computed from first principles when $C$ is \emph{rational}, i.e., its generators can be chosen in $\mathbb{Z}^n$.
A closed cone $C$ is \emph{unimodular} if its generators form a basis of $\mathbb{Z}^n$; for unimodular cones, which is all we will need in what follows, we have the following simple lemma (for much more general results, see, e.g., \cite[Chapter 3]{ccd}).

\begin{lemma}\label{conegenfctlemma}
Suppose $C = \sum_{ j=1 }^{ k } \mathbb{R}_{ \ge 0 } \v_j + \sum_{ i=k+1 }^n \mathbb{R}_{ >0 } \v_i$ is a unimodular cone in $\mathbb{R}^n$ generated by $\v_1, \dots, \v_n \in \mathbb{Z}^n$.
Then
\[
  \sigma_C (z_1, \dots, z_n) = \frac{ \prod_{i=k+1}^n {\mathbf z}^{\v_i} }{ \prod_{ j=1 }^n \left( 1 - {\mathbf z}^{ \v_j } \right) } \, .
\]
\end{lemma}

\section{Unimodular Cones}

Recall from Theorem \ref{higherdiffthm} that
\[
  P_r = \left\{ \lambda : \, \sum_{ j=0 }^t (-1)^j \binom t j \lambda_{ k+j } \ge 0 \ \text{ \rm for } \ k \ge 1, \ 1 \le t \le r \right\}
\]
(where we set undefined $\lambda_j$'s zero).
Let
\[
  P_r^n := \left\{ (\lambda_1, \dots , \lambda_n) \in \mathbb{Z}^n : \, \sum_{ j=0 }^t (-1)^j \binom t j \lambda_{ k+j } \ge 0 \ \text{ \rm for } \ 1 \le k \le n, \ 1 \le t \le r \right\}
\]
consist of all partitions in $P_r$ with at most $n$ parts.
As a warm-up example we will compute the (full) generating function of $P_r^n$:

\begin{theorem}\label{higherdiffthmgeneralized}
\begin{align*}
  &f_{ P_r^n } (z_1, \dots, z_n) = \\
  &\frac{ 1 }{ \left( 1 - z_1 \right) \left( 1 - z_1^r z_2  \right) \left( 1 - z_1^{ \binom{ r+1 }{ r-1 } } z_2^r z_3 \right) \left( 1 - z_1^{ \binom{ r+2 }{ r-1 } } z_2^{ \binom{ r+1 }{ r-1 } } z_3^r z_4 \right) \cdots \left( 1 - z_1^{ \binom{ r+n-2 }{ r-1 } } z_2^{ \binom{ r+n-3 }{ r-1 } } \cdots z_{ n-1 }^r z_n \right) } \, .
\end{align*}
\end{theorem}

\noindent
Note that Theorem \ref{higherdiffthm} follows upon setting $z_1 = \dots = z_n = q$, using the identity
\[
  \binom{ r+j-2 }{ r-1 } + \binom{ r+j-3 }{ r-1 } + \dots + r + 1 = \binom{ r+j-1 }{ r } \, ,
\]
and taking $n \to \infty$.

\begin{proof}
It is easy to see that the inequalities
\[
  \sum_{ j=0 }^t (-1)^j \binom t j \lambda_{ k+j } \ge 0 \ \text{ \rm for } \ 1 \le k \le n, \ 1 \le t \le r \, ,
\]
which define $P_r^n$, are implied by the inequalities for $t=r$.
Thus the cone containing $P_r^n$ as its integer lattice points is
\begin{align*}
  K &:= \left\{ (x_1, \dots , x_n) \in \mathbb{R}^n : \, \sum_{ j=0 }^r (-1)^j \binom r j x_{ k+j } \ge 0 \ \text{ \rm for } \ 1 \le k \le n \right\} \\
    &= \left\{
  \begin{bmatrix}
  1 & r & \binom{ r+1 }{ r-1 } & \binom{ r+2 }{ r-1 } & \cdots & \binom{ r+n-2 }{ r-1 } \\
  0 & 1 & r & \binom{ r+1 }{ r-1 } & \cdots & \binom{ r+n-3 }{ r-1 } \\
  0 & 0 & 1 & r & \cdots & \binom{ r+n-4 }{ r-1 } \\
  \vdots & & \ddots & \ddots & \ddots &\vdots  \ \\
  0 & & & 0 & 1 & r \\
  0 &  & \cdots & & 0 &1 \
  \end{bmatrix}
  {\mathbf y} : \, y_1, \dots, y_n \ge 0 \right\}
\end{align*}
(whose generators we can compute, e.g., with the help of Lemma \ref{linalglemma}).
Thus $K$ is unimodular and, by Lemma~\ref{conegenfctlemma},
\begin{align*}
  &\sigma_K (z_1, \dots, z_n) = \\
  &\frac{ 1 }{ \left( 1 - z_1 \right) \left( 1 - z_1^r z_2  \right) \left( 1 - z_1^{ \binom{ r+1 }{ r-1 } } z_2^r z_3 \right) \left( 1 - z_1^{ \binom{ r+2 }{ r-1 } } z_2^{ \binom{ r+1 }{ r-1 } } z_3^r z_4 \right) \cdots \left( 1 - z_1^{ \binom{ r+n-2 }{ r-1 } } z_2^{ \binom{ r+n-3 }{ r-1 } } \cdots z_{ n-1 }^r z_n \right) } \, .
\end{align*}
\end{proof}

The idea behind this approach towards Theorem \ref{higherdiffthm} can be found, in disguised form, in \cite{corteelsavagewilf} and \cite{pakpartitionprocams}.
See also \cite{canfieldcorteelhitczenko,corteelsavageinequalities} for bijective approaches to Theorem \ref{higherdiffthm} and its asymptotic consequences.
We included this proof here in the interest of a self-contained exposition and also because none of \cite{andrewsPA2,corteelsavagewilf,pakpartitionprocams} contains a full generating function version of (analogues of) Theorem~\ref{higherdiffthm}.

\section{Differences of two cones}

They key idea behind the proof of Theorem~\ref{ngonpartitionthm} is to observe that the non-simplicial cone
\[
  K := \left\{ (x_1, \dots , x_n) \in \mathbb{R}^n : \, x_n \ge \cdots \ge x_1 > 0 \ \text{ and } \ x_1 + \cdots + x_{n-1} > x_n \right\} ,
\]
whose integer lattice points form Andrews--Paule--Riese's set $\tau$ of $n$-gon partitions, can be written as a difference $K = K_1 \setminus K_2$ of two simplicial cones.
Specifically, set
\begin{align*}
  K_1 &:= \left\{ (x_1, \dots , x_n) \in \mathbb{R}^n : \, x_n \ge \cdots \ge x_1 > 0 \right\} \\
    &= \left\{
  \begin{bmatrix}
  1&0&0&\cdots&\ 0 \quad &0 \ \\
  1&1&0&\cdots&\ 0 \quad &0 \ \\
  1&1&1&\cdots&\ 0 \quad &0 \ \\
  \vdots&\vdots&\vdots&\ddots&\ \vdots \quad &\vdots  \ \\
  1&1&1&\cdots&\ 1 \quad &0 \ \\
  1&1&1&\cdots&\ 1 \quad &1 \
  \end{bmatrix}
  {\mathbf y} : \begin{array}{l}
    y_1 > 0 \, , \\
    y_2, \dots, y_n \ge 0
  \end{array} \right\}
\end{align*}
and
\begin{align*}
  K_2 &:= \left\{ (x_1, \dots , x_n) \in \mathbb{R}^n : \, x_n \ge \cdots \ge x_1 > 0 \ \text{ and } \ x_1 + \cdots + x_{n-1} \le x_n \right\} \\
  &= \left\{
  \begin{bmatrix}
  1&0&0&\cdots&\ 0 \quad &0 \ \\
  1&1&0&\cdots&\ 0 \quad &0 \ \\
  1&1&1&\cdots&\ 0 \quad &0 \ \\
  \vdots&\vdots&\vdots&\ddots&\ \vdots \quad &\vdots  \ \\
  1&1&1&\cdots&\ 1 \quad &0 \ \\
  n-1&n-2&n-3&\cdots&\ 1 \quad &1 \
  \end{bmatrix}
  {\mathbf y} : \begin{array}{l}
    y_1 > 0 \, , \\
    y_2, \dots, y_n \ge 0
  \end{array} \right\}
\end{align*}
(whose generators we can compute, e.g., with the help of Lemma \ref{linalglemma}).
One can see immediately from the generator matrices that both $K_1$ and $K_2$ are unimodular.
(In a geometric sense, this is suggested by the form of the identity in Theorem \ref{ngonpartitionthm}. A similar simplification-through-taking-differences phenomenon is described in the fifth ``guideline" of Corteel, Lee, and Savage \cite{corteelleesavage5guidelines}, which inspired our proof.)
By Lemma~\ref{conegenfctlemma}
\[
  \sigma_{K_1} (z_1, \dots, z_n) = \frac{ z_1 \cdots z_n }{ (1 - z_n) (1 - z_{ n-1 } z_n) \cdots (1 - z_1 \cdots z_n)}
\]
and
\begin{align*}
  &\sigma_{K_2} (z_1, \dots, z_n) \\
  &\qquad = \frac{ z_1 \cdots z_{ n-1 } z_n^{ n-1 } }{ \left(1 - z_1 \cdots z_{ n-1 } z_n^{ n-1 } \right) \left(1 - z_2 \cdots z_{ n-1 } z_n^{ n-2 } \right) \left(1 - z_3 \cdots z_{ n-1 } z_n^{ n-3 } \right) \cdots \left(1 - z_{ n-1 } z_n \right) \left(1 - z_n \right) } \\
  &\qquad = \frac{ Z_1 Z_n^{n-2} }{ (1-Z_n) (1- Z_{n-1}) (1 - Z_{n-2} Z_n) (1 - Z_{n-3} Z_n^2) \cdots (1 - Z_1 Z_n^{n-2}) } \, ,
\end{align*}
and the identity $\sigma_K (z_1, \dots, z_n) = \sigma_{K_1} (z_1, \dots, z_n) - \sigma_{K_2} (z_1, \dots, z_n)$ completes the proof.
\hfil $\Box$

\section{Differences of multiple cones}

The ``cone behind" Theorem~\ref{hermitethm} is
\[
  K := \left\{ (x_1, \dots , x_n) \in \mathbb{R}_{ >0 }^n : \, x_j \le x_1 + \dots + \widehat{ x_j } + \dots + x_n \ \text{ for all } \ 1 \le j \le n \right\} ;
\]
Theorem \ref{hermitethm} follows from the following result upon setting $z_1 = \dots = z_n = q$.

\begin{theorem}\label{fullhermitethm}
\[
  \sigma_K (z_1, \dots, z_n) = \frac{ z_1 \cdots z_n }{ (1-z_1) \cdots (1-z_n) } - \sum_{ k=1 }^n \frac{ z_1 \cdots z_{ k-1 } z_k^n z_{ k+1 } \cdots z_n }{ (1-z_k) \prod_{ {j=1} \atop {j \ne k} }^n (1 - z_k z_j) } \, .
\]
\end{theorem}

\begin{proof}
Let ${\mathbf e}_j$ denote the $j$th unit vector in $\mathbb{R}^n$.
Observe that the non-simplicial cone $K$ is expressible as a difference
$
  K = O \setminus \bigcup_{ k=1 }^n C_k \, ,
$
where $O := \sum_{ j=1 }^n \mathbb{R}_{ >0 } \, {\mathbf e}_j$ and $C_k$ is the cone
\begin{align*}
C_k &:= \left\{ (x_1, \dots , x_n) \in \mathbb{R}_{ >0 }^n : \, x_k > x_1 + \dots + \widehat{ x_j } + \dots + x_n  \right\} \\
&= \mathbb{R}_{ >0 } \, {\mathbf e}_k + \sum_{ {j=1} \atop {j \ne k} }^n \mathbb{R}_{ >0 } \left( {\mathbf e}_j + {\mathbf e}_k \right) \\
\end{align*}
Note that if $i\neq j$, then $C_i\cap C_j=\emptyset$.
Thus, the closure of $K$ is ``almost'' the positive orthant $O$, except that we have to exclude points in $O$ that can only be written as a linear combination that requires a single ${\mathbf e}_k$ (as opposed to a linear combination of the vectors ${\mathbf e}_j + {\mathbf e}_k$).
(A similar simplification-through-taking-differences phenomenon appeared in the original proof of Theorem \ref{hermitethm}.)
In generating-function terms, this set difference gives, by Lemma~\ref{conegenfctlemma},
\begin{align*}
  \sigma_K (z_1, \dots, z_n)
  &= \sigma_O (z_1, \dots, z_n) - \sum_{ k=1 }^n \sigma_{ C_k } (z_1, \dots, z_n) \\
  &= \frac{ z_1 \cdots z_n }{ (1-z_1) \cdots (1-z_n) } - \sum_{ k=1 }^n \frac{ z_1 \cdots z_{ k-1 } z_k^n z_{ k+1 } \cdots z_n }{ (1-z_k) \prod_{ {j=1} \atop {j \ne k} }^n (1 - z_k z_j) } \, . \qedhere
\end{align*}
\end{proof}

Three remarks on this theorem are in order.
First, as already mentioned, Xin \cite[Example 6.1]{xinmacmahon} previously computed a different full-generating-function related to Theorem \ref{hermitethm}; Xin's generating function handles non-negative, rather than positive, $k$-gon partitions.
Second, the cone $K$ is related to the second hypersimplex, a well-known object in geometric combinatorics (see Section~\ref{hypersimplices} for more details).

Third, $K$ is a suitable candidate for the ``symmetrically constrained" approach in \cite{beckgesselleesavage}; however, one should expect that this approach would give a different form for the generating function $\sigma_K (z_1, \dots, z_n)$ from the one given in Theorem~\ref{fullhermitethm}.
The symmetrically constrained approach produces a triangulation of the cone $K$ that is invariant under permutation of the standard basis vectors in $\mathbb{R}^n$, and then uses this triangulation to express $\sigma_K(z_1,\dots,z_n)$ as a positive sum of rational generating functions for these cones (after some geometric shifting).
The terms in this sum will all have $\frac{1}{1-z_1 z_2 \cdots z_n}$ as a factor, as each of the simplicial cones in the triangulation of $K$ will have the all-ones vector as a ray generator; this will clearly produce a different form from that in Theorem~\ref{fullhermitethm}.

\section{Cayley Compositions}\label{cayleycomp}

A \emph{Cayley composition} is a composition $\lambda=\left( \lambda_1, \dots, \lambda_{j-1} \right)$ that satisfies $1 \le \lambda_1 \le 2$ and $1 \le \lambda_{ i+1 } \le 2 \lambda_i$ for $1 \le i \le j-2$.
Thus, the Cayley compositions with $j-1$ parts are precisely the integer points in
\[
  C_{j} := \left\{ (\lambda_1, \dots , \lambda_{j-1}) \in \mathbb{Z}_{ >0 }^{j-1} : \, \lambda_1 \leq 2 \ \text{ \rm and } \ \lambda_i \leq 2\lambda_{i-1} \ \text{ \rm for all } \ 2 \le i \le j-1 \right\} \, .
\]
Our apparent shift in indexing maintains continuity between our statements and \cite{andrewspauleriesestrehlPA5}, where Cayley compositions always begin with a $\lambda_0=1$ part.
Let $f_{C_j} \left( z_1, \dots, z_{j-1} \right)$ be the generating function for $C_j$.
The following theorem is quite surprising.

\begin{theorem}[Andrews--Paule--Riese--Strehl \cite{andrewspauleriesestrehlPA5}] \label{cayley}
Let
\[
  C_{j} := \left\{ (\lambda_1, \dots , \lambda_{j-1}) \in \mathbb{Z}_{ >0 }^{j-1} : \, \lambda_1 \leq 2 \ \text{ \rm and } \ \lambda_i \leq 2\lambda_{i-1} \ \text{ \rm for all } \ 2 \le i \le j-1 \right\} \, .
\]
Then for $j\geq 2$,
\begin{align*}
f_{C_{j}}(1,1,\ldots,1,q) = & \sum_{h=1}^{j-2}\frac{b_{j-h-1}(-1)^{h-1}q^{2^h-1}}{(1-q)(1-q^2)(1-q^4)\cdots(1-q^{2^{h-1}})} \\
&
 +  \frac{(-1)^jq^{2^{j-1}-1}(1-q^{2^{j-1}})}{(1-q)(1-q^2)(1-q^4)\cdots(1-q^{2^{j-2}})}
\end{align*}
where $b_k$ is the coefficient of $q^{2^k-1}$ in the power series expansion of
\[
\frac{1}{1-q}\prod_{m=0}^\infty\frac{1}{1-q^{2^m}}
\, .
\]
\end{theorem}

Theorem~\ref{cayley} is derived as a consequence of the following recurrence relation obtained via MacMahon's $\Omega$ calculus.

\begin{theorem}[Andrews--Paule--Riese--Strehl \cite{andrewspauleriesestrehlPA5}]\label{recurrence}
\[
f_{C_{j}} \left( z_1, \dots, z_{j-1} \right) = \frac{ z_{j-1} }{ 1 - z_{j-1} } \left( f_{ C_{j-1} } \left( z_1, \dots, z_{ j-2 } \right) - f_{ C_{j-1} } \left( z_1, \dots, z_{ j-3 }, z_{ j-2 } z_{j-1}^2 \right) \right) .
\]
\end{theorem}

Once this formula is obtained, the proof of Theorem~\ref{cayley} in \cite{andrewspauleriesestrehlPA5} proceeds by repeatedly iterating the recurrence, specialized to $f_{C_j}(1,\ldots,1,q)$.
The final step is to argue that the sum of rational functions in Theorem~\ref{cayley}, as analytic functions, must exhibit cancellation.
We remark that Corteel, Lee, and Savage \cite[Section 3]{corteelleesavage5guidelines} gave an alternative proof of Theorem~\ref{recurrence}.

Via geometry, we can shed light on the initial recurrence relation from three perspectives.
First, we recognize that the recurrence reflects expressing $C_j$ as a difference of two subspaces of $\mathbb{R}^{j-1}$ defined by linear constraints.

\begin{proof}[First proof of Theorem~\ref{recurrence}]
As a subspace of $\mathbb{R}^{j-1}$, $C_j=K_{1,j}\setminus K_{2,j}$ where
\[K_{1,j}:=\left\{ (x_1, \dots , x_{j-1}) \in \mathbb{R}^{j-1} : \, 1\leq x_1 \le 2, \ 1\leq x_{ i+1 } \le 2 x_i \ \text{ for } \ 1 \le i \le j-3, \text{ and } 1\leq x_{j-1} \right\}\]
and
\[K_{2,j}:=\left\{ (x_1, \dots , x_{j-1}) \in \mathbb{R}^{j-1} : \, 1\leq x_1 \le 2, \ x_{ i+1 } \le 2 x_i \, \text{ for } \, 1 \le i \le j-3, \ x_{j-1} > 2 x_{ j-2 } \right\} \, .\]
If we distribute the leading multiplier in the right-hand side of the recurrence for $f_{C_j}$, the first term is the generating function of $K_{1,j}$, as there are no restrictions on the size of $x_{j-1}$.
On the other hand, the integer points ${\mathbf m} \in K_{2,j}$ are precisely those in $K_{1,j}$ satisfying $x_{j-1}>2x_{j-2}$, which is equivalent to the condition that ${\mathbf z}^{\mathbf m}$ be divisible by $z_{j-2}z_{j-1}^2$.
The second term of the recurrence records precisely these integer points.
\end{proof}

Our second proof amounts to a simple observation regarding the integer-point transform of $C_j$.

\begin{proof}[Second proof of Theorem~\ref{recurrence}]
Since for any $\lambda\in C_j\cap \mathbb{Z}^{j-1}$ we have $1\leq \lambda_{j-1}\leq 2\lambda_{j-2}$,
\begin{align*}
f_{C_{j}}(z_1,\ldots,z_{j-1})
&= \sum_{\lambda\in C_{j}\cap \mathbb{Z}^{j-1}}{\mathbf z}^\lambda \\
&= \sum_{\lambda\in C_{j-1}\cap \mathbb{Z}^{j-2}} {\mathbf z}^\lambda(z_{j-1}+z_{j-1}^2+\cdots+z_{j-1}^{2\lambda_{j-2}}) \\
&= z_{j-1} \sum_{\lambda \in C_{j-1}\cap \mathbb{Z}^{j-2}} {\mathbf z}^\lambda \frac{1-z_{j-1}^{2\lambda_{j-2}}}{1-z_{j-1}} \\
&= \frac{z_{j-1}}{1-z_{j-1}}\sum_{\lambda \in C_{j-1}\cap \mathbb{Z}^{j-2}} {\mathbf z}^\lambda - {\mathbf z}^\lambda z_{j-1}^{2\lambda_{j-2}} \\
&= \frac{z_{j-1}}{1-z_{j-1}} \left( f_{C_{j-1}}(z_1,\ldots,z_{j-1})-f_{C_{j-1}}(z_1,\ldots,z_{j-3},z_{j-2}z_{j-1}^2) \right) . \qedhere
\end{align*}
\end{proof}

Following their statement of Theorem~\ref{cayley}, the authors of \cite{andrewspauleriesestrehlPA5} make the following comment:
\begin{quote}
It hardly needs to be pointed out that [this formula] is a surprising representation of a polynomial.
Indeed, the right-hand side does not look like a polynomial at all.
\end{quote}
Such a statement suggests that Brion's formula \cite{brion} for rational polytopes is lurking in the background; our third proof of Theorem~\ref{recurrence} is based on this formula.
Given a rational convex polytope $P$, we first define the \emph{tangent cone} at a vertex $\v$ of $P$ to be
\[
  T_P(\v) := \left\{ \v+\alpha ({\mathbf p}-\v) : \, \alpha \in \mathbb{R}_{ \ge 0 } , \, {\mathbf p} \in P \right\} .
\]
\begin{theorem}[Brion]
Suppose $P$ is a rational convex polytope.
Then we have the following identity of rational generating functions:
\[
\sigma_{P}({\mathbf z})=\sum_{\v \text{ \rm a vertex of } P}\sigma_{T_P(\v)}({\mathbf z}) \, .
\]
\end{theorem}
Note that the sum on the right-hand side is a sum of rational functions, while the left-hand side yields a polynomial.

\begin{proof}[Third proof of Theorem~\ref{recurrence}]
To interpret the recurrence as a consequence of Brion's formula, we first assume that the $f_{C_{j-1}}$'s are expressed in the form of the right-hand side of Brion's formula, i.e., as a sum of integer-point transforms of the tangent cones at the vertices of $C_{j-1}$.
We next rewrite the recurrence as
\[
f_{C_{j}} \left( z_1, \dots, z_{j-1} \right) = \frac{ z_{j-1} }{ 1 - z_{j-1} } \, f_{ C_{j-1} } \left( z_1, \dots, z_{ j-2 } \right) + \frac{1}{1 - z_{j-1}^{-1}} \, f_{ C_{j-1} } \left( z_1, \dots, z_{ j-3 }, z_{ j-2 } z_{j-1}^2 \right) .
\]
The polytope $C_j$ is a combinatorial cube; this can be easily seen by induction on $j$ after observing that in $C_{j-1}\times \mathbb{R}$ the hyperplanes $x_{j-1}=1$ and $x_{j-1}=2x_{j-2}$ do not intersect.
Thus, the tangent cones for vertices of $C_j$ can be expressed in terms of the tangent cones for vertices of $C_{j-1}$.
Given a vertex $\v=\{v_1,\ldots,v_{j-2}\}$ of $C_{j-1}$, the two vertices of $C_j$ obtained from $\v$ are $(\v,1)$ and $(\v,2v_{j-2})$.
For the vertex $(\v,1)$ in $C_j$, it is immediate that
\[
\sigma_{T_{C_{j-1}}((\v,1))}({\mathbf z})=\frac{1}{1-z_{j-1}}\, \sigma_{T_{C_{j-2}}(\v)}({\mathbf z}) \, .
\]
Our proof will be complete after we show that for the vertex $(\v,2v_{j-2})$ in $C_j$,
\[
\sigma_{T_{C_{j-1}}((\v,2v_{j-2}))}({\mathbf z})=\frac{1}{1-z_{j-1}^{-1}}\, \sigma_{T_{C_{j-2}}(\v)}(z_1,\ldots,z_{j-3},z_{j-2}z_{j-1}^2) \, .
\]
This follows from the fact that the edges in $C_j$ emanating from $(\v,2v_{j-2})$ terminate in the vertex $(\v,1)$ and in the vertices $({\mathbf w},2w_{j-2})$ for vertices ${\mathbf w}$ of $C_{j-1}$ that are connected to $\v$ by an edge in $C_{j-1}$.
Thus, Theorem~\ref{recurrence} follows from Brion's formula and induction.
\end{proof}

There is an interesting remark about Theorem~\ref{cayley} and Brion's formula; while one might hope that the expression in Theorem~\ref{cayley} is obtained by directly specializing Brion's formula to $z_1=\cdots=z_{j-2}=1$ and $z_{j-1}=q$, this is not the case.
This specialization is not actually possible, as some of the rational functions for tangent cones in $C_j$ have denominators that lack a $z_{j-1}$ variable, and hence this specialization would require evaluating rational functions at poles.
The authors of \cite{andrewspauleriesestrehlPA5} use the recurrence in Theorem~\ref{recurrence} in a more subtle way, in that they first specialize the recurrence to
\[
f_{C_{j}} \left( 1, \dots, 1,q \right) = \frac{ q }{ 1 - q } \left( f_{ C_{j-1} } \left( 1, \dots, 1 \right) - f_{ C_{j-1} } \left( 1, \dots, 1, q^2 \right) \right)
\]
and then iterate the recurrence.
In doing this, they simultaneously use the interpretation of $f_{C_j}({\mathbf z})$ as a polynomial (for the all-ones specialization) and also the interpretation of $f_{C_j}({\mathbf z})$ as a rational function (for the specialization involving $q^2$).
Thus, while Theorem~\ref{cayley} looks similar to a Brion-type result, it is obtained differently.
We remark that by specializing $z_1=\cdots=z_{j-1}=q$ in Brion's formula for $C_j$, one would obtain a representation of the polynomial $f_{C_j}(q,\ldots,q)$ as a sum of rational functions of~$q$.

\section{Directions for further investigation}\label{hypersimplices}

\subsection{Cones over hypersimplices}

We can view the cone $K$ of the previous section as a cone over a ``half-open'' version of the second \emph{hypersimplex}
\[
  \Delta(2,n) := \left\{ (x_1,\ldots,x_n)\in [0,1]^n : \, \sum_{i=1}^nx_i=2 \right\} ,
\]
in the following manner.
The linear inequality $x_j\geq x_1+\cdots +\widehat{x_j}+ \cdots + x_n$ is equivalent to $\frac{\sum_{i=1}^nx_i}{2}\leq x_j$. When $\sum_{i=1}^nx_i=1$, we are considering the ``slice'' of $K$ that is constrained by $0<x_j\leq \frac{1}{2}$ and $\sum_{i=1}^nx_i=1$, which is $\frac{1}{2}$ of $\Delta(2,n)$ with the condition that $0<x_j$ for all $j$.
From this perspective, we can view the $n$-gon compositions of $t$ as
\begin{align*}
  H(t)
  &:= \left\{ (\lambda_1, \dots , \lambda_n) \in \mathbb{Z}_{ \ge 0 }^n :
  \begin{array}{l}
    \lambda_1 + \dots + \lambda_n = t \, , \\
    \lambda_j \le \lambda_1 + \dots + \widehat{ \lambda_j } + \dots + \lambda_n \ \text{ for all } 1 \le j \le n
  \end{array} \right\} \\
  &= \left\{ (\lambda_1, \dots , \lambda_n) \in \mathbb{Z}^n :
  \begin{array}{l}
    \lambda_1 + \dots + \lambda_n = t \, , \\
    0 \le \lambda_j \le \frac t 2 \ \text{ for all } 1 \le j \le n
  \end{array} \right\} .
\end{align*}
The second hypersimplex is a well-studied object; for example, in matroid theory $\Delta(2,n)$ is the matroid basis polytope for the $2$-uniform matroid on $n$ vertices, while in combinatorial commutative algebra $\Delta(2,n)$ is the subject of~\cite[Chapter 9]{sturmfels}.

It would be interesting to consider analogues of Theorem~\ref{hermitethm} for the general case of the $k^{\text{th}}$ hypersimplex $\Delta(k,n):=\{(x_1,\ldots,x_n)\in [0,1]^n : \sum_{i=1}^nx_i=k\}$.
The associated composition counting function has a natural interpretation: in
\[
  \left\{ (\lambda_1, \dots , \lambda_n) \in \mathbb{Z}^n :
  \begin{array}{l}
    \lambda_1 + \dots + \lambda_n = t \, , \\
    0 \le \lambda_j \le \frac t k \ \text{ for all } 1 \le j \le n
  \end{array} \right\}
\]
are all compositions of $t$ whose parts are at most $\frac t k$ (i.e., the parts are not allowed to be too large, where ``too large" depends on~$k$).

\subsection{Cayley polytopes}

We refer to the polytopes $C_j$ from Section~\ref{cayleycomp} as \emph{Cayley polytopes}.
By taking a geometric view of Cayley compositions as integer points in $C_j$, we may shift our focus from combinatorial properties of the integer points to properties of $C_j$ itself.
Recall that the \emph{normalized volume} of $C_j$ is
\[
  \mathrm{Vol}(C_j):=(j-1)! \, \mathrm{vol}(C_j) \, ,
\]
where $\mathrm{vol}(C_j)$ is the Euclidean volume of $C_j$.
Based on experimental data obtained using the software LattE \cite{koeppelatte} and the Online Encyclopedia of Integer Sequences \cite{sloaneonlineseq}, we make the following conjecture:

\begin{conjecture}\label{cayleyconj}
For $j\geq 2$, $\mathrm{Vol}(C_j)$ is equal to the number of labeled connected graphs on $j-1$ vertices.\footnote{
After a preprint of the current article was made public, Matjaz Konvalinka and Igor Pak communicated to us that they resolved Conjecture \ref{cayleyconj} by a direct combinatorial argument.
}
\end{conjecture}

\bibliographystyle{amsplain}


WWE NXT Superstar Dominik Dijakovic has found himself in some hot that led to him deleting his official Twitter account.
This was after posting controversial comments. For some background, Dijakovic expanded his gimmick at Wednesday's NXT TV taping by talking about how his family immigrated to the United States. Thus, he made it known that he wants to win the NXT North American Title from The Velveteen Dream. For those who want to read the spoilers from the taping and what took place with him, you can check that out here.


text:
https://www.nhregister.com/news/article/Middlefield-emergency-management-director-resigns-11545558.php
Middlefield emergency management director resigns
JONETTA BADILLO
Published 12:00 am EDT, Friday, April 6, 2012
MIDDLEFIELD -- First selectman John Brayshaw on Monday announced the resignation of Emergency Management Director Terrie Parmelee.
Brayshaw said Parmelee worked as the emergency management director for the last two to three years and was "excellent" for the job.
"We all know him very well and knew he had all the qualifications," he said Tuesday, a day after the Board of Selectmen meeting. "He was born for the job."
However, Brayshaw said Parmelee recently retired from his full-time job and felt it would be best to resign from his $3,000 stipend-based position with the town.
Though the emergency management position pays very little, it requires a lot from the individual filling it, he said. A director has to be around at all times and take the job seriously, according to Brayshaw.
"[Parmelee] doesn't want those obligations hanging over his head [anymore]," Brayshaw said.
Parmelee could not be reached for comment Thursday.
The entire town is sad to see him go, but, Brayshaw said, for some people a few thousand dollars is "not worth it."
Brian Dumas has taken over since Parmelee stepped down and will be serving as the town's acting emergency director, Brayshaw said.
"He did a good job during the last two storms," he said.
Judge tosses lenient plea deal in 'life threatening' Milford assault


text:
Home > 📰News
Nokia Lumia 920 to come with Wireless Charging & PureView Camera
By Radu Tyrsina Updated Dec 16, 2016
Nokia couldn't release the first Windows Phone 8 smartphone, but maybe that's because they're cooking something interesting in their Finnish kitchen. TheVerge seems to have gotten some inside confirmation that the next model in the Nokia Lumia range, the 920 model, will be released with wireless charging and with the awe-inspiring PureView camera. There are big chances that Nokia Lumia 920 will be Nokia's first Windows Phone 8 device and not Nokia Phi, as we had previously assumed.
Nokia Lumia 920 to have wireless charging and PureView technology
We've already been pleasantly surprised by the wireless charging ability in the Samsung Galaxy S3 (via the wireless charging kit), and we're quite sure the owners were happy to say good-bye to another cord in their life. Nokia Lumia 920 could have some sort of a magnetic strip in its back to be able to support wireless charging. What's different from the feature in the Samsung Galaxy S3 is that Nokia's first Windows Phone 8 handset could make use of the Qi wireless power standard which means it could be compatible with other wireless charging products.
Besides wireless charging, Nokia Lumia 920 is said to come with a PureView camera. We're not sure if it will be as powerful as the 41-megapixel camera in Nokia 808, as word has it that it will rather have a more modest 8-megapixel one, with some of the PureView camera technology built in. Nokia hopes that the PureView branding should set it apart from the rest. But it might actually backfire by smashing up a 5-year effort, if not implemented in fullest. Nokia Lumia 920 won't be a small brick, as there is already confirmation that it will come with a quite big 4.5 inch HD display.
Launch date and place in the mobile world
With 32GB of device storage, 1GB of RAM, and a 1.5GHz dual-core processor and possibly other impressive features, Nokia's first Windows Phone 8 device has all the chances of trying to get back its former glory. Unfortunately for those attending the IFA conference in Berlin, it seems that Nokia Lumia 920 will be launched on September 5th and it will be initially offered only to AT&T customers.
We can safely assume, just like we saw with other Windows Phone 8 smartphones and Windows 8 tablets, that Nokia Lumia 920 will be launched in late October or early November. If the next iPhone will be launched by then, I will be very curious to see how the fight will evolve between manufacturers all over the world, and, in the same time, to see how the mobile operating system will be reshaped. Can Windows Phone 8 really become a serious threat for Android and iOS or it will just have to fight to survive?
Update: Twitter user Evleaks, who was the one behind the first pictures of Nokia Lumia 920, has just revealed two more pictures that we're embedding below. One of them is showing the wireless charging pad and the other one show two models of what seems to be Nokia Lumia 920 & Nokia Lumia 820. Even if these look like rendered images, the final products will have a design that makes us think about toys.
[via] TheVerge
Best Black Friday Phone Deals 2022: iPhones, Pixels, Samsung Galaxy and More
#nokia #Nokia Lumia #smartphones #windows phone 8
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text:
Damiano Sonzogni (n. 1968) – allenatore di calcio ed ex calciatore italiano, di ruolo difensore
 Giuliano Sonzogni (n. 1949) – allenatore di calcio italiano
 Vittorio Sonzogni (1924-2017) –  architetto e urbanista italiano

Pagine correlate
 Sonzogno

redpajama_set_name: RedPajamaWikipedia


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