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\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
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\newtheorem{proposition}[theorem]{Proposition}
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\begin{center}
\vskip 1cm{\LARGE\bf
Nesting Nonpartitions
}
\vskip 1cm
\large
Joshua Marsh and Nathan Williams \\
Department of Mathematical Sciences\\
800 W. Campbell Road \\
Richardson, TX 75080\\
USA \\
\href{mailto:joshuawmarsh@gmail.com}{\tt joshuawmarsh@gmail.com} \\
\href{mailto:nathan.f.williams@gmail.com}{\tt nathan.williams1@utdallas.edu}
\end{center}
\vskip .2 in
\begin{abstract}
Antichains of root posets associated with simple complex Lie algebras
are a well-studied combinatorial object, famously counted by the
Coxeter-Catalan numbers. In this paper, we study chains in root posets
using standard enumerative techniques. The factorizations of the zeta
polynomials of these root posets gives rise to sequences that have
similar numerological properties as the exponents of the Weyl group.
\end{abstract}
\section{Introduction}
Call an interval $[a,b]$ \defn{$n$-integral} if $a$ and $b$ are integers $1 \leq a**=latex,line join=bevel,]
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\caption{The 20 nesting nonpartitions on $n=4$ points, ordered by inclusion.}
\label{fig:ex1}
\end{figure}
It is not difficult to show that the number of nesting nonpartitions on $n+1$ points is given by
\begin{equation}
\tag{\seqnum{A006012}}
\label{eq:a006012}
\sum_{k=0}^n\sum_{i=0}^k \binom{n}{2i}\binom{n-i}{k-i}
=\frac{\left(2+\sqrt{2}\right)^{n} + \left(2-\sqrt{2}\right)^{n}}{2}.
\end{equation}
Define a \defn{$k$-multinesting nonpartition} to be a set of $k$ $n$-integral intervals such that every pair of intervals nests (intervals may now appear with multiplicity). As we show in Section \ref{sec:typea}, the number of $k$-multinesting nonpartitions is given by the compact expression
\begin{equation}
\label{eq:count_generic}
\zeta_{A_{n}}(k)=\prod_{i=1}^{n-1} \frac{2k+i}{i}.
\end{equation}
Nesting nonpartitions on $n+1$ points are naturally interpreted as chains in the root poset of type $A_n$---we will review root systems and root posets in Section \ref{sec:background}. This phrasing allows us to define the \defn{$k$-multinesting nonpartitions} for a general irreducible root system $\Phi$ as the multichains of length $k$ in the restriction of the root poset to the short roots, and state a root-theoretic generalization of Equation \ref{eq:count_generic}.
\begin{theorem}\label{thm:main} Let $\Phi$ be an irreducible crystallographic root system. Then the number of $k$-multinesting nonpartitions is given by
\[\zeta_{\Phi^+_s}(k)=\prod_{i=1}^{g-2} \frac{2k+\db_i}{\db_i},\]
for the sequence of positive integers $\db_1 \leq \db_2 \leq \cdots \leq \db_{g-2}=h-2$ given in Table \ref{fig:table}, where $h$ is the Coxeter number of $\Phi$, and $g$ is the dual Coxeter number of the dual root system $\Phi^{\vee}$.
\end{theorem}
Our proofs are case-by-case, and---although we show in Proposition \ref{prop:num} that the sequences $(\db_i)_{i=1}^{g-2}$ have connections to the numerology of the root system---we have been unable to find a representation-theoretic interpretation of these numbers.
\section{Background}
\label{sec:background}
\subsection{Root systems}
Let $E$ be a finite-dimensional vector space with inner product $\langle \cdot \,, \cdot \rangle$ and let $\alpha, \beta \in E$. The reflection $\sigma_\alpha$ through the hyperplane perpendicular to $\alpha$ sends any vector parallel to $\alpha$ to its negative while leaving any vector perpendicular to $\alpha$ unchanged. By linearity, we see that \[\sigma_\alpha(\beta) = - \frac{\langle \alpha, \beta \rangle}{\langle \alpha, \alpha \rangle} \alpha + \beta - \frac{\langle \alpha, \beta \rangle}{\langle \alpha, \alpha \rangle}\alpha = \beta - 2 \frac{\langle \alpha, \beta \rangle}{\langle \alpha, \alpha \rangle} \alpha.\]
\begin{table}[htbp]
\[\begin{array}{|c|c|} \hline
W & \ec_1,\ec_2,\ldots,\ec_{g-1} \\ \hline \hline
A_n & \begin{array}{cccc} 1 &2 &\ldots & n\end{array} \\ \hline
B_n & \begin{array}{cccc} 1 &3 &\ldots & 2n{-}1 \end{array}\\ \hline
C_n & \begin{array}{cccccc}1 & 3 & 5 & \ldots & 2n{-}3 & 2n{-}1 \\ & 3 & 5 &\ldots & 2n{-}3 & \end{array} \\ \hline
D_n & \begin{array}{cccccc}1 & 3 & 5 &\ldots & 2n{-}5 &2n{-}3 \\& & & n{-}1 & & \\ & 3 & 5 & \ldots & 2n{-}5 &\end{array}\\ \hline
E_6 & \begin{array}{ccccccccc} 1 & & 4 & 5 & & 7 & 8 & & 11 \\ & 3 & & 5 & & 7 & & 9 & \\ & & & & 6 & & & & \end{array}\\ \hline
E_7 & \begin{array}{ccccccccc} 1 & & 5 & 7 & 9 & 11 & 13 & & 17 \\ & 3 & 5 &7 & 9 & 11 & 13 & 15 & \\ & & & 7 & 9 & 11 & & & \end{array} \\ \hline
E_8 & \begin{array}{ccccccccccccccc} 1 & & & 7 & & 11 & 13 & & 17 & 19 & & 23 & & & 29 \\ & 3& 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 21 & 23 & 25 & 27 & \\ & & & & 9 & 11 & 13 & 15 &17 & 19 & 21 & & & & \\ & & & & & & & 15 & & & & & & & \end{array} \\ \hline
F_4 & \begin{array}{cccccc} 1 & & 5 & 7 & & 11 \\ & 3 & 5 &7 & 9 & \end{array} \\ \hline
G_2 & \begin{array}{ccc} 1 & & 5 \\ & 3 & \end{array} \\ \hline
\end{array}\]
\caption{The sequences $(\protect\ec_i)_{i=1}^{g-1}$ for the irreducible crystallographic root systems, arranged to illustrate their symmetry, where $\protect\ec_{i+1}=\protect\db_{i}+1$ for $1 \leq i \leq g-2$ and $\protect\ec_1=1$. The top line for each root system consists of its exponents.}
\label{fig:table}
\end{table}
\begin{definition}
A (crystallographic) \defn{root system} is a finite set $\Phi \subseteq E$ of nonzero vectors spanning $E$ such that for $\alpha, \beta \in \Phi$ we have
\begin{itemize}
\item $\operatorname{span}(\alpha) \cap \Phi = \{\alpha, -\alpha\}$,
\item $\sigma_\alpha(\Phi) = \Phi$, and
\item $2\frac{\langle \alpha, \beta \rangle }{\langle \alpha, \alpha \rangle}$ is an integer.
\end{itemize}
The \defn{Weyl group} $W$ associated with a root system $\Phi$ is the subgroup of $O(E)$ generated by the reflections $\sigma_\alpha$ for $\alpha \in \Phi$.
\end{definition}
The condition that $2\frac{\langle \alpha, \beta \rangle }{\langle \alpha, \alpha \rangle}$ is an integer, is the crystallographic condition---it may be restated as ``$\sigma_\alpha(\beta)$ may be obtained from $\beta$ by adding an integer multiple of $\alpha$.'' A root system is \defn{reducible} if it is the union of two disjoint root systems, each of which spans one of a pair of orthogonal subspaces of $E$. Thus, a reducible root system can be viewed as the union of these two independent root systems. A root system which has no such decomposition is called \defn{irreducible}.
By the well-known classification of irreducible root systems~\cite{Hum90}, there can be at most two different root lengths. If there are two different lengths, we call the corresponding sets of roots the \defn{short roots} and \defn{long roots}. A root system for which there is only one length of roots is called \defn{simply laced}. The root systems $A_n, D_n, E_6, E_7,$ and $E_8$ are simply laced, while $B_n, C_n, F_4,$ and $G_2$ are not.
\begin{example}
\label{ex:roota}
Let $E$ be the $n$-dimensional space of vectors in $\RR^{n+1}$ whose entries sum to zero, and let $e_i$ be the standard basis vectors of $\RR^{n+1}$. Define $\alpha_i = e_i - e_{i+1}$, so that $\Delta=\{ \alpha_1,\alpha_2,\ldots,\alpha_n\}$ is a basis for $E$. This set generates (by reflecting in these roots) the full root system of type $A_n$: \[\Phi_{A_n} = \{e_i-e_j : 1 \leq i,j \leq n+1\}.\] Reflections in the hyperplane orthogonal to $e_i-e_j$ exchange the $i$th and $j$th coordinates, so that the Weyl group associated with the $A_n$ root system is isomorphic to the symmetric group $S_{n+1}$ (acting on $E$ by permuting coordinates).
\end{example}
The root system $A_n$ is often the first example of a root system, and combinatorics done in this context is often referred to as ``Type $A$,'' in contrast to combinatorics done in the context of general root systems. More precisely, many classical combinatorial objects can be interpreted as arising from the symmetric group or $\Phi_{A_n}$ in some way, and---once phrased in that language---can often be generalized by passing to other root systems. In this spirit, this paper interprets nesting nonpartitions in the context of type $A$ root systems and extends their definition to other root systems.
\subsection{Root posets}
By choosing a generic hyperplane in $E$ that does not contain any root, we can divide a root system $\Phi$ into positive and negative roots. The roots closest to the hyperplane form a basis for $E$, which allows us to partially order the positive roots.
\begin{definition}
A \defn{simple system} is a subset $\Delta = \{\alpha_1, \alpha_2, \dots, \alpha_n\} \subseteq \Phi$ such that
\begin{itemize}
\item the elements of $\Delta$ are linearly independent, and
\item every element of $\Phi$ can be written as a linear combination of elements of $\Delta$, either with all nonnegative or all nonpositive coefficients.
\end{itemize}
\end{definition}
\begin{definition}
A set of \defn{positive roots} is a choice of roots $\Phi^+ \subseteq \Phi$ such that
\begin{itemize}
\item for $\alpha \in \Phi$, exactly one of $\alpha$ and $-\alpha$ is in $\Phi^+$, and
\item if a root is the sum of two roots in $\Phi^+$, it is also in $\Phi^+$.
\end{itemize}
\end{definition}
We call the roots in the simple system the \defn{simple roots}.
The notions of positive roots and simple systems are closely related. It is easy to see that any choice of a simple system determines a set of positive roots; the roots which can be obtained from a nonnegative linear combination of simple roots may be designated as positive roots, and the rest negative. Conversely, any choice of positive roots uniquely determines a simple system~\cite[Section 1.3]{Hum90}. In particular, this means that a simple system exists, since we may choose a total order on $E$ that is compatible with the vector space operations (for example, the lexicographic order), and this order determines a positive set and therefore a simple system. Moreover, all simple systems are essentially the same, differing only by the action of the Weyl group~\cite[Section 1.4]{Hum90}. More precisely, for any two simple systems $\Delta$ and $\Delta'$, there is some $w \in W,$ the Weyl group of the root system, so that $\Delta' = w \Delta$. Since $W \subseteq O(E)$, the simple systems are orthogonal transformations of each other and have the same geometry. Thus, this choice is immaterial and we will make convenient choices of simple systems.
The choice of positive roots, along with vector space operations, allows us to define a partial order on the positive roots.
\begin{definition}
\label{def:short_root}
The \defn{root poset} of an irreducible crystallographic root system $\Phi$ is the poset on the positive roots $\Phi^+$ defined by $\alpha\leq \beta$ if $\beta-\alpha$ is a nonnegative sum of positive roots. The \defn{short root poset} $\Phi^+_s$ is the restriction of $\Phi^+$ to the short roots when there are two different root lengths, and the root poset $\Phi^+$ otherwise.
\end{definition}
\begin{example}
Continuing Example \ref{ex:roota}, the set $\Delta = \{\alpha_1, \alpha_2, \dots, \alpha_n\}$ is a convenient choice of simple roots (and will be assumed to be the set of simple roots for $A_n$ for the rest of the paper). Writing $\alpha_{i,j} = e_i - e_j$, the corresponding positive roots are \[\Phi^+_{A_n} = \{ \alpha_{i,j} \mid 1\leq i < j \leq n+1 \}.\] The difference between two positive roots, $\alpha_{i,j} - \alpha_{k, l}$, is itself a nonnegative sum of simple roots when $ i \leq k < l \leq j$.
The $A_5$ root poset is illustrated in Figure \ref{fig:typea}.
\end{example}
\begin{figure}[htbp]
\begin{center}
\includegraphics{rootposeta.png} \hspace{1in}
\includegraphics{rootposetd.png}
\end{center}
\caption{{\it Left}: the $A_5$ positive root poset. {\it Right}: the $D_6$ positive root poset.}
\label{fig:typea}
\end{figure}
\subsection{Numerology}
Associated with a root system $\Phi$ are two important invariants, the sequence of \defn{degrees} $(d_i)_{i=1}^n$ and the sequence of \defn{exponents} $(e_i)_{i=1}^n$. It turns out that we can easily define both using the positive root poset (although this is not how they were defined historically). A positive root $\alpha$ can be expressed as a nonnegative sum of simple roots: $\alpha=\sum_{i=1}^n a_i \alpha_i$. Define the \defn{height} of a positive root $\alpha$ to be $\sum_{i=1}^n a_i$---the number of simple roots that must be added to get $\alpha$ (this is the rank of $\alpha$ in the positive root poset).
If $k_i$ is the number of positive roots of height $i$, we always have \[k_1 \geq k_2 \geq \dots \geq k_{h-1} = 1.\] This defines a partition of $|\Phi|/2$, the number of positive roots. From this partition, we define the sequence of exponents (in descending order) by the \emph{dual} partition $e_n \geq e_{n-1} \geq \dots \geq e_1$ of $|\Phi^+|$ with $k_1 = n$ parts. We define the \defn{Coxeter number} by $h=e_n+1=|\Phi| / n$, the degrees by $d_i=e_i+1$, and the \defn{dual Coxeter number of the dual root system $g$} as one plus the height of the highest short root. In fact, it turns out that these numbers satisfy
\begin{equation}
\label{eq:exponents}
\sum_{i=1}^{n} e_i = nh/2 \hfill \text{ and }\hfill
e_i+e_{n+1-i} = h.
\end{equation}
As explained in~\cite[Chapter 3]{Hum90}, there are two other algebraic settings in which the exponents and degrees appear.
The first comes from the invariant theory of the Weyl group. Let $\Phi$ be an irreducible crystallographic root system spanning an $n$-dimensional vector space $E$, and let $W \subseteq O(E)$ be its Weyl group, generated by the reflections of $\Phi$. We associate $E$ with $\RR^n$, and $W$ acts on the polynomial ring $S = \RR[x_1,x_2, \dots, x_n]$ in a natural way by transforming the vectors $(x_1,x_2, \dots, x_n)$. Let $R$ be the ring of polynomials invariant under the action of $W$ and $f_1,f_2, \dots, f_n$ be a set of homogeneous, algebraically independent polynomials that generate $R$. Then the sequence $d_1 \leq d_2 \leq \cdots \leq d_n$, where $d_i$ is the degree of $f_i$, is independent of the choice of generating polynomials (up to reordering). These degrees are therefore an important invariant of the root system.
The second comes from the eigenvalues of certain elements in the Weyl group. Since any $w \in W$ is an orthogonal transformation of finite order, its eigenvalues must be roots of unity. Fix a choice of simple roots $\Delta = \{\alpha_1,\alpha_2, \dots, \alpha_n\}$ with corresponding reflections $\sigma_1,\sigma_2, \dots, \sigma_n$. Any element that is the product of all simple reflections (in any order) is called a \defn{Coxeter element}, and all such elements are conjugate, with the same order $h$. The eigenvalues of a Coxeter element are powers of $\zeta$, where $\zeta$ is a primitive $h$th root of unity. If $(\zeta^{e_i})_{i=1}^n$ are the eigenvalues of a Coxeter element, the sequence $e_1 \leq e_2 \leq \dots \leq e_n$ recovers the exponents of $W$.
\subsection{Posets}
Recall that a \defn{chain} in a poset $P$ is a sequence of elements $p_1**