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\newcommand{\LV}{\mbox{$L$}}
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\newcommand{\DC}{\mbox{${\cal C}_{\cal D}^{n+1}$}}
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\begin{document}



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\title{Facets of the cone up to rank 7}

\author{Margaret M. Bayer\thanks{This research was supported by University of
Kansas General Research allocation \#3552.}\ \ and 
        G\'abor Hetyei\thanks{On leave from the
        Alfr\'{e}d R\'{e}nyi Institute.
        Partially supported by Hungarian National Foundation for
        Scientific Research grant no. F 032325.}\\
        Department of Mathematics\\
        University of Kansas\\
        Lawrence KS 66045-2142\\
}

\date{April 28, 2000}

\maketitle

\noindent This note is available on the Worldwide Web at the address
\begin{center}
{\tt http://www.math.ku.edu/\~{}bayer/euler/index.html}
\end{center}
and it provides supplemental information to our paper
{\em ``Flag vectors of Eulerian partially ordered sets'',} to appear in
the European Journal of Combinatorics. 

\section{Description of notation}


\begin{description}
\item[Convolution of chain operators:] $f^m_S f^n_T\df f^{m+n}_{S\cup
\{m\}\cup (T+m)}$. \hfill\break
See Appendix B of our paper. 
\item[Sparse $f$ basis:] A set $S\subseteq \{1,2,\ldots,n\}$ is {\em
sparse} if it does not contain two consecutive integers, and it does not
contain $n$. For the vector space of chain operators $\langle f^{n+1}_S\::\:
S\subseteq \{1,2,\ldots,n\}\rangle$ acting on Eulerian posets of rank $n+1$,
the set $\{f^{n+1}_S\::\: S\subseteq \{1,2,\ldots,n\}, \mbox{$S$
sparse}\}$ forms a basis. This was shown in: 

M.\ M.\ Bayer and L. J.\ Billera, Generalized
Dehn--Sommerville relations for polytopes, spheres and Eulerian partially
ordered sets, {\it Invent. Math.} {\bf 79} (1985), 143--157.

\item[$\LV$-vector:] $\di\LV^{n+1}_S\df (-1)^{n-|S|} \sum _{T\supseteq
[1,n]\setminus S} \left(-{1\over 2}\right)^{|T|} f^{n+1}_T$. 
Equivalently $\di f^{n+1}_S=2^{|S|} \sum _{T\subseteq
[1,n]\setminus S} \LV^{n+1}_T$. \hfill\break
See Definition 2 of our paper.

\item[The cones:] \EC\ is the smallest closed convex cone containing the
flag vectors of all Eulerian posets.

\DC\ is the smallest closed convex cone containing the
flag vectors of the horizontal doubles of all half-Eulerian posets.
(See Section~2 of our paper.)
\end{description}

\section{From PORTA output to Eulerian cone}

Let $\cal V$ be a set of vectors contained in the cone of
half-Eulerian flag $f$-vectors, and let $\cone({\cal V})$ be the smallest
convex cone containing $\cal V$.
Consider the affine transformation $\phi$ that multiplies each 
$f_S$-entry of a vector by $2^{|S|}$, and 
divides the coefficient of $f_S$ in an inequality by $2^{|S|}$.
Let ${\cal K}^{n+1}$ be the image of $\cone({\cal V})$ under $\phi$.
Then ${\cal K}^{n+1}$ is contained in 
the closed cone ${\cal C}_{\cal D}^{n+1}$ of flag $f$-vectors of horizontal 
doubles of half-Eulerian posets, which is contained
in the Eulerian cone ${\cal C}_{\cal E}^{n+1}$.

For each rank $n+1\le 7$ we created a finite set $\cal V$ in the cone of
half-Eulerian flag $f$-vectors, and used PORTA to generate $\cone({\cal V})$.
In this note we list the facet inequalities for the corresponding cone
${\cal K}^{n+1}$ ($n\le 6$).
We check that these inequalities are valid for all Eulerian posets
(not just for doubled half-Eulerian posets), and this tells us that 
${\cal C}_{\cal E}^{n+1} \subseteq {\cal K}^{n+1}$ for rank at most 7.  Thus
${\cal K}^{n+1} = {\cal C}_{\cal D}^{n+1} = {\cal C}_{\cal E}^{n+1} $,
so we have in fact determined the closed cone of Eulerian flag $f$-vectors 
for rank $n+1\le 7$.

\section{Inequalities in the sparse $f$ basis}
\label{s_sparse}

The inequalities listed are equivalent up to a constant factor to the
inequalities encoded in the corresponding PORTA output files.
The second number in brackets corresponds to the number of the equivalent
inequality in the $\LV$-basis. Boldfaced inequalities cannot be written
as convolutions of lower-rank inequalities.

\begin{description}

\item[{\bf Rank 1:}] 
$$
\begin{array}{lrr}
{\mathbf (F 1)}&{\mathbf f^1_{\es}\geq 0}& {\mathbf (L 1)}\\ 
\end{array}
$$

\item[{\bf Rank 2:}] 
$$
\begin{array}{lrr}
{\mathrm (F 1)} & f^2_\es\geq 0 & {\mathrm (L 1)}\\
\end{array}
$$
$\mathrm{(F 1)}$ is equivalent to ${1\over 2}f^2_1={1\over 2}f^1_{\es} f^1_{\es}\geq 0$. 

\item[{\bf Rank 3:}] 
$$
\begin{array}{lrr}
{\mathbf (F 1)}&{\mathbf f^3_\es\geq 0}& {\mathbf (L 2)}\\
{\mathbf (F 2)}& {\mathbf -2f^3_\es+f^3_1\geq 0}& {\mathbf (L 1)}\\
\end{array}
$$



\item[{\bf Rank 4:}] 
$$
\begin{array}{lrr}
{\mathbf (F 1)}&{\mathbf f^4_\es \geq 0}&{\mathbf (L 3)}\\    
{\mathrm (F 2)}& -f^4_1+f^4_2 \geq 0 & {\mathrm (L 2)} \\   
{\mathrm (F 3)}& -2f^4_\es+f^4_1 \geq 0 & {\mathrm (L 1)}\\   
\end{array}
$$
{\bf\noindent Equivalent forms of reducible or non-evident inequalities:}
\vspace*{-0.3cm}
\begin{itemize}
\itemsep=-3pt
\item[${\mathrm (F 2)}$]: ${1\over 2}(f^4_{12}-2f^4_1)={1\over
2}f^1_\es(f^3_1-2f^3_\es)\geq 0$.  
\item[${\mathrm (F 3)}$]:
$f^4_2-f^4_3={1\over 2}(f^4_{13}-2f^4_3)
={1\over 2}(f^3_1-2f^3_{\es})f^1_{\es}\geq 0$.   
\end{itemize}

\item[{\bf Rank 5:}] 
$$
\begin{array}{lrr}
{\mathbf (F 1)}&{\mathbf  f^5_\es \geq 0}&{\mathbf (L 6)}\\    
{\mathrm (F 2)}&  -2f^5_3+f^5_{13} \geq 0& {\mathrm (L 3)}\\    
{\mathrm (F 3)}&  -2f^5_2   +f^5_{13} \geq 0& {\mathrm (L 1)}\\    
{\mathrm (F 4)}&  -f^5_1+f^5_2       \geq 0& {\mathrm (L 2)} \\    
{\mathbf (F 5)}&{\mathbf -2f^5_\es+f^5_1 \geq 0}&{\mathbf (L 5)}\\    
{\mathbf (F 6)}&{\mathbf -2f^5_\es+f^5_1-f^5_2+f^5_3 \geq 0}&{\mathbf (L 4)}\\    
\end{array}
$$
{\bf\noindent Equivalent forms of reducible or non-evident inequalities:}
\vspace*{-0.3cm}
\begin{itemize}
\itemsep=-3pt
\item[${\mathrm (F 2)}$]: $(f^3_1-2f^3_{\es})f^2_{\es}={1\over
2}(f^3_1-2f^3_{\es})f^1_{\es}f^1_{\es}\geq 0$.  
\item[${\mathrm (F 3)}$]:
${1\over 2}(f^5_{123}-4f^5_2)={1\over 2}(f^5_{123}-2f^5_{12})={1\over 2}
f^1_{\es}f^1_{\es}(f^3_1-2f^3_{\es})\geq 0$. 
\item[${\mathrm (F 4)}$]: 
${1\over 2}(f^5_{12}-2f^5_1)={1\over 2}f^1_\es (f^4_1-2f^4_\es)={1\over
4}f^1_\es (f^3_1-2 f^3_\es) f^1_\es \geq 0$. 
\item[${\mathbf (F 6)}$]: ${\mathbf f^5_4-2f^5_{\es}\geq 0}$. 
\end{itemize}

\item[{\bf Rank 6:}] 
$$
\begin{array}{lrr}
{\mathbf (F 1)}&{\mathbf f^6_\es                      \geq 0} & {\mathbf (L 10)}  \\    
{\mathrm (F 2)}& -f^6_{14}+f^6_{24} \geq 0 &   {\mathrm (L 2)}\\    
{\mathrm (F 3)}& -2f^6_3+f^6_{13}          \geq 0 & {\mathrm (L 6)}  \\    
{\mathrm (F 4)}& -2f^6_2   +f^6_{13}          \geq 0 & {\mathrm (L 1)}  \\    
{\mathrm (F 5)}& -f^6_1+f^6_2                \geq 0 &  {\mathrm (L 5)} \\    
{\mathbf (F 6)}&{\mathbf -2f^6_\es+f^6_1 \geq 0} &{\mathbf (L 8)}  \\    
{\mathrm (F 7)}& 2f^6_3-f^6_{13}-2f^6_4+f^6_{14}    \geq 0 & {\mathrm (L 9)}  \\    
{\mathrm (F 8)}& -2f^6_3   +2f^6_4-f^6_{14}+f^6_{24} \geq 0 & {\mathrm (L 3)}  \\    
{\mathbf (F 9)}&{\mathbf -f^6_1+f^6_2-f^6_3 +f^6_4\geq 0} &{\mathbf (L 7)}\\
{\mathrm (F 10)}& -2f^6_\es+f^6_1-f^6_2+f^6_3\geq 0 & {\mathrm (L 4)}\\ 
\end{array}
$$
{\bf\noindent Equivalent forms of reducible or non-evident inequalities:}
\vspace*{-0.3cm}
\begin{itemize}
\itemsep=-3pt
\item[${\mathrm (F 2)}$]: $(f^4_1-f^4_2)f^2_\es={1\over 4}f^1_\es(f^3_1-2f^3_\es)f^1_{\es}f^1_{\es} \geq 0$.
\item[${\mathrm (F 3)}$]: $(f^3_1-f^3_{\es})f^3_{\es}\geq 0$.  
\item[${\mathrm (F 4)}$]: ${1\over 2}(-2f^6_{12}+f^6_{123})
={1\over 2}f^1_\es f^1_\es(2f^4_\es-f^4_1)
={1\over 4}f^1_\es f^1_\es (f^3_1-2f^3_{\es})f^1_{\es}\geq 0$. 
\item[${\mathrm (F 5)}$]: ${1\over 2}(f^6_{12}-2f^6_1)
={1\over 2}f^1_{\es}(f^5_1-2f^5_{\es})$.
\item[${\mathrm (F 7)}$]: $2f^6_3-f^6_{23}+f^6_{24}-f^6_{34}
={1\over 2}(4f^6_3-2f^6_{23}+f^6_{234}-2f^6_{34})
={1\over 2}(f^3_2-2f^3_{\es})(f^3_1-2f^3_{\es})$\break
$={1\over 2}(f^3_1-2f^3_{\es})(f^3_1-2f^3_{\es})
\geq 0$.
\item[${\mathrm (F 8)}$]: $f^6_{34}-2f^6_3=f^3_{\es}(f^3_1-2f^3_{\es})\geq 0$.
\item[${\mathbf (F 9)}$]: ${\mathbf f^6_5-2f^6_{\es}\geq 0}$.
\item[${\mathrm (F10)}$]: $f^6_4-f^6_5={1\over 2}(f^6_{45}-2f^6_5)
={1\over 2}(f^5_4-2f^5_{\es})f^1_{\es}\geq 0$.
\end{itemize}

\item[{\bf Rank 7:}] 
$$
\begin{array}{lrr}
{\mathbf (F 1)}&{\mathbf f^7_\es \geq 0} &{\mathbf (L 23)}\\    
{\mathrm (F 2)}& -2f^7_{25} +f^7_{135} \geq 0& {\mathrm (L 1)}\\    
{\mathrm (F 3)}& -f^7_{14}+f^7_{24}\geq 0& {\mathrm (L 5)}\\    
{\mathrm (F 4)}& -2f^7_3+f^7_{13} \geq 0& {\mathrm (L 10)}\\    
{\mathrm (F 5)}& -2f^7_2 +f^7_{13} \geq 0& {\mathrm (L 3)}\\    
{\mathrm (F 6)}& -f^7_1+f^7_2 \geq 0& {\mathrm (L 9)}\\    
{\mathbf (F 7)}&{\mathbf -2f^7_\es+f^7_1\geq 0}&{\mathbf (L 13)}\\    
{\mathbf (F 8)}& {\mathbf -2f^7_\es+f^7_3\geq 0}&{\mathbf (L 15)}\\    
{\mathbf (F 9)}&{\mathbf -2f^7_\es+f^7_4\geq 0}&{\mathbf (L 14)} \\    
{\mathrm (F 10)}& 2f^7_3-f^7_{13}-2f^7_4+f^7_{14}\geq 0& {\mathrm (L 16)}\\     
{\mathrm (F 11)}& 4f^7_4-2f^7_{14}-2f^7_{35}+f^7_{135} \geq 0& {\mathrm (L 17)}\\    
{\mathrm (F 12)}& f^7_{14}-f^7_{24}-f^7_{15}+f^7_{25}\geq 0& {\mathrm (L 18)}\\    
{\mathrm (F 13)}& -2f^7_5+f^7_{15}-f^7_{25}+f^7_{35} \geq 0& {\mathrm (L 4)} \\    
{\mathrm (F 14)}& -2f^7_4+f^7_{15}-f^7_{25}+f^7_{35}\geq 0& {\mathrm (L 7)}\\    
{\mathrm (F 15)}& -2f^7_3+2f^7_4-f^7_{14}+f^7_{24} \geq 0& {\mathrm (L 2)}\\    
{\mathbf (F 16)} &{\mathbf -2 f^7_2+2f^7_3-2f^7_5+f^7_{25}\geq 0} &{\mathbf (L 22)}\\    
{\mathrm (F 17)}& -2f^7_2+f^7_{13}-f^7_{14}+f^7_{15}\geq 0& {\mathrm (L 6)}\\    
{\mathbf (F 18)} &{\mathbf -2 f^7_2+2f^7_4-2f^7_5+f^7_{25}\geq 0} &{\mathbf (L 21)}\\    
{\mathrm (F 19)}& -f^7_1+f^7_2-f^7_3 +f^7_4 \geq 0& {\mathrm (L 8)}\\    
{\mathrm (F 20)}& -2f^7_3+f^7_{13}+2f^7_4-f^7_{14}-2f^7_5+f^7_{15} \geq 0& {\mathrm (L 11)}\\    
{\mathbf (F 21)} &{\mathbf -2 f^7_1+f^7_{13}+2f^7_4-f^7_{14}-2f^7_5+
        f^7_{15} \geq 0} &{\mathbf (L 20)}\\    
{\mathbf (F 22)}
	&{\mathbf -2f^7_\es+f^7_1-f^7_2+f^7_3   -f^7_4 +f^7_5 \geq 0}
	&{\mathbf (L 12)}\\    
{\mathbf (F 23)} &{\mathbf
        -2f^7_1-2f^7_3+f^7_{13}+4f^7_4-f^7_{14}-2f^7_5+f^7_{15} \leq 0}
	&{\mathbf (L 19)}\\    
\end{array}
$$
\newpage
{\bf\noindent Equivalent forms of reducible or non-evident inequalities:}
\vspace*{-0.3cm}
\begin{itemize}
\itemsep=-3pt
\item[${\mathrm (F 2)}$]: $(-2f^5_2+f^5_{13})f^2_\es
={1\over 4}f^1_{\es}f^1_{\es}(f^3_1-2f^3_{\es})f^1_{\es}f^1_{\es}
\geq 0$. 
\item[${\mathrm (F 3)}$]: $(f^4_2-f^4_1)f^3_\es
={1\over 2}f^1_{\es}(f^3_1-2f^3_{\es})f^3_{\es}\geq 0$. 
\item[${\mathrm (F 4)}$]: $(f^3_1-2f^3_{\es})f^4_{\es}\geq 0$.
\item[${\mathrm (F 5)}$]: ${1\over 2}(-2f^7_{12}   +f^7_{123})
={1\over 2}f^1_\es f^1_\es (f^5_1-2f^5_{\es})\geq 0$.
\item[${\mathrm (F 6)}$] ${1\over 2}(-2f^7_1+f^7_{12})
={1\over 2}f^1_{\es}(f^6_1-2f^6_{\es})\geq 0$.
\item[${\mathrm (F 10)}$]: $2f^7_3-f^7_{23}+f^7_{24}-f^7_{34}
={1\over 2}(4f^7_3-2f^7_{23}+f^7_{234}-2f^7_{34})
={1\over 2}(f^3_1-2f^3_\es)(f^4_1-2f^4_\es)$\break
$={1\over 4}(f^3_1-2f^3_\es)(f^3_1-2f^3_{\es})f^1_{\es}\geq 0$.
\item[${\mathrm (F 11)}$]:
$-2f^7_{24}+2f^7_{34}-2f^7_{35}+f^7_{135}
={1\over 2}(-2f^7_{134}+4f^7_{34}-2f^7_{345}+f^7_{1345})
={1\over 2}(f^3_1-2f^3_\es)f^1_\es(f^3_1-2f^3_\es)\geq 0$. 
\item[${\mathrm (F 12)}$]: 
$2f^7_4-f^7_{34}+f^7_{35}-f^7_{45}
={1\over 2}(4f^7_4-2f^7_{34}+f^7_{345}-2f^7_{45})
={1\over 2}(f^4_3-2f^4_\es)(f^3_1-2f^3_{\es})$\break
$
={1\over 2}(f^4_2-f^4_1)(f^3_1-2f^3_{\es})
={1\over 4}f^1_{\es}(f^3_2-2f^3_{\es})(f^3_1-2f^3_{\es})
={1\over 4}f^1_{\es}(f^3_1-2f^3_{\es})(f^3_1-2f^3_{\es})
\geq 0$.
\item[${\mathrm (F 13)}$]: $f^7_{45}-2f^7_{5}
={1\over 2}(f^5_4-2f^5_\es)f^1_\es f^1_\es\geq 0$.
\item[${\mathrm (F 14)}$]:
$f^7_{45}-2f^7_4=f^4_{\es}(f^3_1-2f^3_{\es})\geq 0$.  
\item[${\mathrm (F 15)}$]: $f^7_{34}-2f^7_3 =f^3_\es
(f^4_1-2f^4_\es)
={1\over 2} f^3_\es (f^3_1-2f^3_{\es})f^1_{\es}\geq 0$.   
\item[${\mathrm (F 17)}$]: $f^7_{16}-2f^7_1=
f^1_{\es}(f^6_5-2f^6_{\es})\geq 0$.
\item[${\mathrm (F 19)}$]: $f^7_5-f^7_6
={1\over 2}(f^7_{56}-2f^7_6)
={1\over 2}(f^6_5-2f^6_{\es})f^1_{\es}\geq 0$.
\item[${\mathrm (F 20)}$]: $f^7_{16}-2f^7_6
=(f^6_1-2f^6_\es)f^1_\es\geq 0$.
\item[${\mathbf (F 21)}$]: ${\mathbf f^7_{16}-2f^7_1-2f^7_6+2f^7_3\geq 0}$.
\item[${\mathbf (F 22)}$]: ${\mathbf f^7_6-2f^7_\es \geq 0}$.
\item[${\mathbf (F 23)}$]: ${\mathbf f^7_{16}-2f^7_1-2f^7_6+2f^7_4\geq 0}$.
\end{itemize}


\end{description}



\section{Inequalities in the $\LV$-basis}
\label{s_lv}

{\bf\noindent Note:} Since $\di f^{n+1}_S=2^{|S|}\sum _{T\subseteq
[1,n]\setminus S} \LV^{n+1}_T$, and here we need to consider only the
nonzero $\LV^{n+1}_T$'s, it is easy to convert an $f$-formula into an
$\LV$-formula. The other way may be more difficult.
The labels ${\mathrm (F i)}$ refer to the numbering in the sparse $f$
basis which is given in section \ref{s_sparse}. 
Boldfaced inequalities cannot be written
as convolutions of lower-rank inequalities.


\begin{description}
\item[{\bf Rank 1:}] 
$$
\begin{array}{lrr}
{\mathbf (L 1)}&{\mathbf -\LV^1_{\es}\leq 0}& {\mathbf (F 1)}\\ 
\end{array}
$$

\item[{\bf Rank 2:}] 
$$
\begin{array}{lrr}
{\mathrm (L 1)} & -\LV^2_{\es}\leq 0 & {\mathrm (F 1)}\\
\end{array}
$$

\item[{\bf Rank 3:}] 
$$
\begin{array}{lrr}
{\mathbf (L 1)}& {\mathbf \LV^3_{12}\leq
0}& {\mathbf (F 2)}\\
{\mathbf (L 2)}&{\mathbf -\LV^3_{\es}-\LV^3_{12}\leq 0}& {\mathbf (F 1)}\\
\end{array}
$$

\item[{\bf Rank 4:}] 
$$
\begin{array}{lrr}
{\mathrm (L 1)}& \LV^4_{12}=\LV^3_{12}\LV^1_{\es}\leq 0 & {\mathrm (F 3)}\\
{\mathrm (L 2)}& \LV^4_{23}=\LV^1_{\es} \LV^3_{12}\leq 0 & {\mathrm (F 2)}\\ 
{\mathbf (L 3)} 
&{\mathbf -\LV^4_\es-\LV^4_{12}-\LV^4_{23}\leq 0} &{\mathbf (F 1)}\\
\end{array}
$$

\item[{\bf Rank 5:}] 
$$
\begin{array}{lrr}
{\mathrm (L 1)}&          \LV^5_{34}    \leq 0& {\mathrm (F 3)}\\    
{\mathrm (L 2)}&       \LV^5_{23}       \leq 0& {\mathrm (F 4)}\\    
{\mathrm (L 3)}&    \LV^5_{12}          \leq 0& {\mathrm (F 2)}\\    
{\mathbf (L 4)}&{\mathbf           \LV^5_{34}+\LV^5_{1234} \leq 0}
	&{\mathbf (F 6)}\\    
{\mathbf (L 5)}&{\mathbf    \LV^5_{12}      +\LV^5_{1234}} \leq 0
	&{\mathbf (F 5)}\\    
{\mathbf (L 6)}&\mathbf
{ -\LV^5_{\es}-\LV^5_{12}-\LV^5_{23}-\LV^5_{34}-\LV^5_{1234} \leq 0}
	&{\mathbf (F 1)}\\
\end{array}
$$

\item[{\bf Rank 6:}] 
$$
\begin{array}{lrr}
{\mathrm (L 1)}&          \LV^6_{34}             \leq 0& {\mathrm (F 4)}\\    
{\mathrm (L 2)}&       \LV^6_{23}                \leq 0& {\mathrm (F 2)}\\    
{\mathrm (L 3)}&                \LV^6_{45}+\LV^6_{1245}  \leq 0& {\mathrm (F 8)}\\    
{\mathrm (L 4)}&          \LV^6_{34}+\LV^6_{1234}          \leq 0& {\mathrm (F 10)}\\    
{\mathrm (L 5)}&       \LV^6_{23}            +\LV^6_{2345} \leq 0& {\mathrm (F 5)}\\    
{\mathrm (L 6)}&   \LV^6_{12}            +\LV^6_{1245}    \leq 0& {\mathrm (F 3)}\\    
\mathbf{(L 7)}&\mathbf
{                \LV^6_{45}+\LV^6_{1245}+\LV^6_{2345} \leq 0}& (F 9)\\    
\mathbf{(L 8)}&\mathbf{\LV^6_{12}      +\LV^6_{1234}   +\LV^6_{1245}
\leq 0}& {\mathbf (F 6)}\\    
{\mathrm (L 9)}&                   -\LV^6_{1245}    \leq 0& {\mathrm (F 7)}\\    
\mathbf{(L
10)}&\mathbf{-\LV^6_\es-\LV^6_{12}-\LV^6_{23}-\LV^6_{34}-\LV^6_{1234}-\LV^6_{45}-\LV^6_{1245}-\LV^6_{2345}}
\leq 0& {\mathbf (F 10)}\\    
\end{array}
$$

\item[{\bf Rank 7:}] 
$$
\begin{array}{lrr}
{\mathrm (L 1)}& \LV^7_{34}\leq 0& {\mathrm (F 2)}\\    
{\mathrm (L 2)}& \LV^7_{45}+\LV^7_{1245} \leq 0& {\mathrm (F 15)} \\    
{\mathrm (L 3)}& \LV^7_{34}+\LV^7_{3456}     \leq 0& {\mathrm (F 5)}\\    
{\mathrm (L 4)}& \LV^7_{34}+\LV^7_{1234}     \leq 0& {\mathrm (F 13)}\\    
{\mathrm (L 5)}& \LV^7_{23}+\LV^7_{2356}         \leq 0& {\mathrm (F 3)}\\    
{\mathrm (L 6)}& \LV^7_{56}    +\LV^7_{2356}+\LV^7_{3456} \leq 0& {\mathrm (F 17)}\\    
{\mathrm (L 7)}& \LV^7_{56}+\LV^7_{1256}+\LV^7_{2356} \leq 0& {\mathrm (F 14)}\\    
{\mathrm (L 8)}& \LV^7_{45}+\LV^7_{1245}+\LV^7_{2345} \leq 0& {\mathrm (F 19)}\\    
{\mathrm (L 9)}& \LV^7_{23}+\LV^7_{2345}       +\LV^7_{2356}         \leq 0& {\mathrm (F 6)}\\    
{\mathrm (L 10)}& \LV^7_{12}+\LV^7_{1245}+\LV^7_{1256}  \leq 0& {\mathrm (F 4)}\\    
{\mathrm (L 11)}& \LV^7_{12}+\LV^7_{1234}+\LV^7_{1245}  \leq 0& {\mathrm (F 20)}\\    
{\mathbf (L 12)}&{\mathbf \LV^7_{56}+\LV^7_{1256}+\LV^7_{2356}+\LV^7_{3456}
       +\LV^7_{123456}\leq 0}& {\mathbf (F 22)}\\ 
{\mathbf (L 13)}&{\mathbf \LV^7_{12}+\LV^7_{1234}+\LV^7_{1245}+\LV^7_{1256}
       +\LV^7_{123456}\leq 0}& {\mathbf (F 7)}\\ 
{\mathbf (L 14)}&{\mathbf \LV^7_{34}+\LV^7_{1234}+\LV^7_{45}
	+\LV^7_{1245}+\LV^7_{2345}+\LV^7_{3456}+\LV^7_{123456}\leq 0}
	& {\mathbf (F 9)}\\      
{\mathbf (L 15)}& 
{\mathbf \LV^7_{23}+\LV^7_{34}+\LV^7_{1234}+\LV^7_{2345}+\LV^7_{2356}
    +\LV^7_{3456}+\LV^7_{123456}\leq 0}& {\mathbf (F 8)}\\    
{\mathrm (L 16)}& -\LV^7_{1245} \leq 0& {\mathrm (F 10)}\\    
{\mathrm (L 17)}& -\LV^7_{1256} \leq 0& {\mathrm (F 11)}\\    
{\mathrm (L 18)}& -\LV^7_{2356} \leq 0& {\mathrm (F 12)}\\    
{\mathbf (L 19)}&{\mathbf \LV^7_{34} + \LV^7_{1234} + \LV^7_{45} +
\LV^7_{1245} + \LV^7_{2345} - \LV^7_{1256} + \LV^7_{3456} \leq 0}& {\mathbf (F 23)}\\    
{\mathbf (L 20)}&{\mathbf \LV^7_{23} + \LV^7_{34} + \LV^7_{1234} +
\LV^7_{2345} - \LV^7_{1256} + \LV^7_{2356} + \LV^7_{3456}\leq 0}& {\mathbf (F 21)}\\    
{\mathbf (L 21)}&{\mathbf \LV^7_{34}+\LV^7_{1234}+
\LV^7_{45}- \LV^7_{1256} -\LV^7_{2356}+\LV^7_{3456}\leq 0}&{\mathbf (F 18)}\\ 
{\mathbf (L 22)}&{\mathbf \LV^7_{23} + \LV^7_{34} + \LV^7_{1234} 
-\LV^7_{1245}-\LV^7_{1256}+ \LV^7_{3456}\leq 0}& {\mathbf (F 16)}\\    
{\mathbf (L 23)}&{\mathbf
-\LV^7_\es-\LV^7_{12}-\LV^7_{23}-\LV^7_{34}-\LV^7_{1234}-\LV^7_{45}\ \ \
}& {\mathbf (F 1)}\\
  &{\mathbf  -\LV^7_{1245}-\LV^7_{2345}-\LV^7_{56}-\LV^7_{1256}-\LV^7_{2356}
       -\LV^7_{3456}-\LV^7_{123456} \leq 0}& \\    
\end{array}
$$
\end{description}

\end{document}