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\title {Subword complexes via triangulations of root polytopes}
\author[\initial{L.} Escobar]{\firstname{Laura} \lastname{Escobar}}
\address{University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 W. Green Street, Urbana, IL 61801, USA}
\email{lescobar@illinois.edu}
\thanks{Escobar was partially supported by an AMS-Simons travel grant. M{\'e}sz\'aros was partially supported by a National Science Foundation Grant (DMS 1501059).}
\author[\initial{K.} M{\'e}sz\'aros]{\firstname{Karola} \lastname{M{\'e}sz\'aros}}
\address{Cornell University, Department of Mathematics, 310 Malott Hall, Ithaca, NY 14850, USA}
\email{karola@math.cornell.edu}
\keywords{subword complex, pipedream, triangulation, root polytope}
\subjclass{52B20, 05E45}
\begin{abstract}
Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of $\beta$-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We can also write the volume and Ehrhart series of root polytopes in terms of $\beta$-Grothendieck polynomials.
\end{abstract}
\begin{document}
\maketitle
\section{Introduction}
In this paper we provide geometric realizations of pipe dream complexes $\PD(\pi)$ of permutations $\pi=1\pi'$, where $\pi'$ is a dominant permutation on $2,3,\dots, n$ as well as the subword complexes that are the cores of the pipe dream complexes $\PD(\pi)$. We realize $\PD(\pi)$ as (repeated cones of) regular triangulations of the root polytopes $\P(T(\pi))$.
Subword complexes are simplicial complexes introduced by Knutson and Miller in~\cite{subword,annals} to illustrate the combinatorics of Schubert polynomials and determinantal ideals. Since the appearance of Knutson's and Miller's work in there has been a flurry of research into the geometric realization of subword complexes with progress in realizing families of spherical subword complexes:~\cite{reviewedfansub,cesar,subwordcluster,escobar,assoc,abrick,SerranoStump,stump}. This paper is the first to succeed in realizing a family of subword complexes which are homeomorphic to balls.
Subword complexes were first shown to relate to triangulations of root polytopes by M{\'e}sz\'aros in~\cite{groth}, where the author gives a geometric realization of the pipe dream complex of $[1,n,n-1,\dots,2]$ and whose work served as the stepping stone for the present project. In the papers~\cite{root1, h-poly1, prod, h-poly2} M{\'e}sz\'aros studied triangulations of root polytopes that we utilize in this work (some of the mentioned papers are in the language of flow polytopes, but in view of~\cite[Section~4]{groth} some of their content can also be understood in the language of root polytopes).
The main theorem of this paper is the following, which has several interesting consequences explored in the paper. For the definitions needed for this theorem see the later sections.
\begin{theo}\label{thm:intro}
Let $\pi=1\pi' \in S_n$, where $\pi'$ is dominant. Let $\mathcal{C}^2(\pi)$ be the core of $\PD(\pi)$ coned over twice. The canonical triangulation of the root polytope $\P(T(\pi))$ (which is a regular triangulation) is a geometric realization of $\mathcal{C}^2(\pi)$.
\end{theo}
The roadmap of this paper is as follows. In Sections~\ref{sec:pipe} and~\ref{sec:root} we explain the necessary background about subword complexes and root polytopes, respectively. In Section~\ref{sec:constr} we prove a geometric realizations of pipe dream complexes $\PD(\pi)$ of permutations $\pi=1\pi'$, where $\pi'$ is a dominant permutation on $2,3,\dots, n$, via triangulations of root polytopes $\P(T(\pi))$. In Section~\ref{sec:groth} we use the previous result to show that $\beta$-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes while in Section~\ref{sec:cor} we show how to express the volume and Ehrhart series of root polytopes in terms of Grothendieck polynomials. Appendix~\ref{sec:unique} is devoted to proving a certain uniqueness property of the reduced form in the subdivision algebra that we used in Section~\ref{sec:groth}.
\section{Background on pipe dream complexes}\label{sec:pipe}
We let $S_n$ denote the set of permutations of size $n$.
\begin{defi}
The \emph{(Rothe) diagram} of a permutation $\pi\in S_n$ is the collection of boxes $D(\pi)=\{(\pi_j,i): i\pi_j\}$. It can be visualized by considering the boxes left in the $n\times n$ grid after we cross out the boxes appearing south and east of each 1 in the permutation matrix for $\pi$.
\end{defi}
\begin{figure}[h]
\begin{tikzpicture}
\draw (0,0)--(2,0)--(2,2)--(0,2)--(0,0);
\draw[] (2,1.25) -- (1.75,1.25) node {$\bullet$} -- (1.75,0);
\draw[] (2,1.75) -- (0.75,1.75) node {$\bullet$} -- (0.75,0);
\draw[] (2,.75) -- (1.25,.75) node {$\bullet$} -- (1.25,0);
\draw[] (2,.25) -- (.25,.25) node {$\bullet$} -- (.25,0);
\draw (0,.5)--(.5,.5)--(.5,2);
\draw (0,1)--(.5,1);
\draw (0,1.5)--(.5,1.5);
\draw (1,1)--(1.5,1)--(1.5,1.5)--(1,1.5)--(1,1);
\end{tikzpicture}\caption{The diagram for $\pi=[4132]$.}
\end{figure}
Notice that no two permutations can give the same diagram. We will consider permutations of the form $\pi=1\pi'$ where $\pi'$ is a \emph{dominant permutation} of $\{2,\dots,n\}$, i.e., the diagram of $\pi$ is a partition with north-west most box at position $(2,2)$. Dominant permutations can be equivalently defined as the $132$-avoiding permutations, and there are Catalan many for fixed size. Our convention is to encode the partition by the number of boxes in each column.
\begin{figure}[h]
\begin{tikzpicture}
\draw (0,0)--(3,0)--(3,3)--(0,3)--(0,0);
\draw[] (3,2.75) -- (0.25,2.75) node {$\bullet$} -- (0.25,0);
\draw[] (3,0.25) -- (0.75,0.25) node {$\bullet$} -- (0.75,0);
\draw[] (3,1.25) -- (1.25,1.25) node {$\bullet$} -- (1.25,0);
\draw[] (3,2.25) -- (1.75,2.25) node {$\bullet$} -- (1.75,0);
\draw[] (3,1.75) -- (2,1.75) node {$\bullet$} -- (2,0);
\draw[] (3,0.75) -- (2.25,0.75) node {$\bullet$} -- (2.25,0);
\draw (0.5,0.5)--(1,0.5)--(1,2.5)--(0.5,2.5)--(0.5,0.5);
\draw (0.5,1)--(1,1);
\draw (0.5,1.5)--(1.5,1.5);
\draw (0.5,2)--(1.5,2);
\draw (1,2.5)--(1.5,2.5)--(1.5,1.5);
\end{tikzpicture}\caption{The diagram for $\pi=[164235]$ which corresponds to $\lambda=(4,2)$}
\end{figure}
\begin{defi}
A \emph{pipe dream} for $\pi\in S_n$ is a tiling of an $n\times n$ matrix with two tiles, crosses $\textcross$ and elbows $\textelbow$, such that
\begin{enumerate}
\item all tiles in the weak south-east triangle of the $n\times n$ matrix are elbows, and
\item if we write $1,2,\dots, n$ on the left and follow the strands (ignoring second crossings among the same strands) they come out on the top and read $\pi$.
\end{enumerate}
A pipe dream is \emph{reduced} if no two strands cross twice.
\end{defi}
\begin{figure}[h]
\begingroup
\arraycolsep=5pt
$\begin{array}{ccccc} &\perm1{}&\perm4{}&\perm2{}&\perm3{}\\[0.5em]
\petit1\!\!& \jr & \jr & \jr & \je \\[0.2em]
\petit2\!\!& \+ & \jr & \je &\\[0.2em]
\petit3\!\!& \+ & \je & &\\[0.2em]
\petit4\!\!& \je & & &\\[0.2em]
\end{array}
$
\endgroup
\caption{A reduced pipe dream for $\pi=[1423]$.}\label{fig:redpd1423}
\end{figure}
\begin{defi}
The \emph{pipe dream complex} $\PD(\pi)$ of a permutation $\pi\in S_n$ is the simplicial complex with vertices given by entries on the northwest triangle of an $n\times n$-matrix and facets given by the elbow positions in the reduced pipe dreams for $\pi$.
\end{defi}
Pipe dream complexes are a special case of the subword complexes defined by Knutson and Miller in~\cite{subword,annals}. We proceed to explain the correspondence. The group $S_n$ is generated by the adjacent transpositions $s_1,\dots,s_{n-1}$, where $s_i$ transposes $i \leftrightarrow i+1$. Let $Q=(q_1,\dots,q_\sizeQ)$ be a word in $\{s_1,\dots,s_{n-1}\}$, i.e., $Q$ is an ordered sequence. A \emph{subword} $J=(r_1,\dots,r_\sizeQ)$ of $Q$ is a word obtained from $Q$ by replacing some of its letters by $-$. There are a total of $2^{|Q|}$ subwords of $Q$. Given a subword $J$, we denote by $Q\setminus J$ the subword with $k$-th entry equal to $-$ if $r_k\neq -$ and equal to $q_k$ otherwise for, $k=1,\dots,\sizeQ$. For example, $J=(s_1,-,s_3,-,s_2)$ is a subword of $Q=(s_1,s_2,s_3,s_1,s_2)$ and $Q\setminus J=(-,s_2,-,s_1,-)$. Given a subword $J$ we denote by $\prod J$ the product of the letters in $J$, from left to right, with $-$ behaving as the identity.
\begin{defi}[\cite{subword,annals}]
Let $Q=(q_1,\dots,q_\sizeQ)$ be a word in $\{s_1,\dots,s_{n-1}\}$ and $\pi\in S_n$. The \emph{subword complex} $\Delta(Q,\pi)$ is the simplicial complex on the vertex set $Q$ whose facets are the subwords $F$ of $Q$ such that the product $\prod(Q\setminus F)$ is a reduced expression for $\pi$.
\end{defi}
In this language, $\PD(\pi)$ is the subword complex $\Delta(Q,\pi)$ corresponding to the triangular word $Q=(s_{n-1},s_{n-2},s_{n-1},\dots,s_1,s_2,\dots,s_{n-1})$ and $\pi$. The correspondence between pipe dreams and subwords is induced by the labeling of the entries in the northwest triangle of an $n\times n$-matrix by adjacent transpositions, as depicted in Figure~\ref{fig:pipex}, and by making a $\textcross$ in a pipe dream correspond to a $-$ in a subword and a $\textelbow$ correspond to the $s_i$ in its entry. In order to go from a pipe dream to a subword, we read the entries in the northwest triangle from left to right starting at the bottom.
\begin{figure}[h]
\begin{tikzpicture}
\draw (0,0)--(0.5,0)--(0.5,0.5)--(1,0.5)--(1,1)--(1.5,1)--(1.5,1.5)--(2,1.5)--(2,2)--(0,2)--(0,0);
\draw[gray] (0,0.5)--(0.5,0.5)--(0.5,2);
\draw[gray] (0,1)--(1,1)--(1,2);
\draw[gray] (0,1.5)--(1.5,1.5)--(1.5,2);
\node at (0.25,0.25) {$s_4$};
\node at (0.75,0.75) {$s_4$};
\node at (1.25,1.25) {$s_4$};
\node at (1.75,1.75) {$s_4$};
\node at (.25,0.75) {$s_3$};
\node at (0.75,1.25) {$s_3$};
\node at (1.25,1.75) {$s_3$};
\node at (0.25,1.25) {$s_2$};
\node at (0.75,1.75) {$s_2$};
\node at (0.25,1.75) {$s_1$};
\node at (2.3,1.75) {$\cdots$};
\node at (0.25, -0.2) {$\vdots$};
\end{tikzpicture}\caption{Labeling of the entries in the northwest triangle by adjacent transpositions.}\label{fig:pipex}
\end{figure}
\begin{defi}\label{def:cone}
Let cone$(\pi)$ be the set of vertices of $\PD(\pi)$ that are in all its facets. We define the \emph{core} of $\pi$, denoted by $\core(\pi)$, to be the simplicial complex obtained by restricting $\PD(\pi)$ to the set of vertices not in cone$(\pi)$.
\end{defi}
Notice that $\PD(\pi)$ is obtained from its core by iteratively coning the simplicial complex $\core(\pi)$ over the vertices in cone$(\pi)$. This is a standard definition for simplicial complexes. In the language of pipe dream complexes, the core is the restriction to the entries in the $n\times n$ matrix that are a cross in some reduced pipe dream for $\pi$. Following the correspondence described in Figure~\ref{fig:pipex}, this restriction induces a subword $Q'$ of the triangular word and so $\core(\pi)$ is the subword complex $\Delta(Q',\pi)$.
\begin{exem}
Figure~\ref{fig:redpd1423} shows a reduced pipe dream for $\pi=[1423]$.
The pipe dream complex $\PD(1423)$ is three dimensional with vertices $(1,1)$, $(1,2)$, $(1,3)$, $(2,1)$, $(2,2)$, and $(3,1)$. Every facet contains the vertices $(1,1)$ and $(1,3)$ since every reduced pipe dream must have elbows on these positions. Therefore, one can recover $\PD(1423)$ from its core, shown in Figure~\ref{fig:core1423}, by coning twice.
\begin{figure}[h]
\begingroup
\arraycolsep=5pt
\begin{tikzpicture}
\draw[thick] (0,0) node {$\bullet$} -- (3,0) node {$\bullet$} -- (6,0) node {$\bullet$} -- (9,0) node {$\bullet$};
\draw (0,.4) node {$(1,2)$};
\draw (3,.4) node {$(2,2)$};
\draw (6,.4) node {$(2,1)$};
\draw (9,.4) node {$(3,1)$};
\draw (1.2,-1.2) node {$\begin{array}{ccccc} &\perm1{}&\perm4{}&\perm2{}&\perm3{}\\[0.5em]
\petit1\!\!& \jr & \jr & \jr & \je \\[0.2em]
\petit2\!\!& \+ & \jr & \je &\\[0.2em]
\petit3\!\!& \+ & \je & &\\[0.2em]
\petit4\!\!& \je & & &\\
\end{array}$};
\draw (4.2,-1.2) node {$\begin{array}{ccccc} &\perm1{}&\perm4{}&\perm2{}&\perm3{}\\[0.5em]
\petit1\!\!& \jr & \+ & \jr & \je \\[0.2em]
\petit2\!\!& \jr & \jr & \je &\\[0.2em]
\petit3\!\!& \+ & \je & &\\[0.2em]
\petit4\!\!& \je & & &\\
\end{array}$};
\draw (7.2,-1.2) node {$\begin{array}{ccccc} &\perm1{}&\perm4{}&\perm2{}&\perm3{}\\[0.5em]
\petit1\!\!& \jr & \+ & \jr & \je \\[0.2em]
\petit2\!\!& \jr & \+ & \je &\\[0.2em]
\petit3\!\!& \jr & \je & &\\[0.2em]
\petit4\!\!& \je & & &\\
\end{array}$};
\end{tikzpicture}
\endgroup
\caption{The core of $\PD(1423)$. The facets are labelled by the reduced pipe dreams for $[1423]$.}\label{fig:core1423}
\end{figure}
\end{exem}
Since we are only considering permutations of the form $1\pi'$ with $\pi'$ a dominant permutation, $\core(1\pi')$ is easy to describe. Given a diagram of a permutation there are two natural reduced pipe dreams for $\pi$, referred to as the \emph{bottom reduced pipe dream of $\pi$} and the \emph{top reduced pipe dream of $\pi$}, one obtained by aligning the diagram to the left and replacing the boxes with crosses and the other one by aligning the diagram up. See Figure~\ref{fig:topbottomrpd}.
\begin{figure}[h]
\begingroup
\arraycolsep=5pt
\centering
\begin{subfigure}[b]{0.35\textwidth}
\centering $
\begin{array}{ccccccc}
&\perm1{}&\perm6{}&\perm4{}&\perm2{}&\perm3{}&\perm5{}\\[0.5em]
\petit1\!\!& \jr & \jr & \jr & \jr & \jr & \je \\[0.2em]
\petit2\!\!& \+ & \+ & \jr & \jr & \je &\\[0.2em]
\petit3\!\!& \+ & \+ & \jr & \je &\\[0.2em]
\petit4\!\!& \+ & \jr & \je &\\[0.2em]
\petit5\!\!& \+ & \je & &\\[0.2em]
\petit6\!\!& \je & & &\\
\end{array}
$
\caption{Aligned left is the bottom reduced pipe dream}
\end{subfigure}
\qquad\quad
\begin{subfigure}[b]{0.4\textwidth}
\centering $
\begin{array}{ccccccc}
&\perm1{}&\perm6{}&\perm4{}&\perm2{}&\perm3{}&\perm5{}\\[0.5em]
\petit1\!\!& \jr & \+ & \+ & \jr & \jr & \je \\[0.2em]
\petit2\!\!& \jr & \+ & \+ & \jr & \je &\\[0.2em]
\petit3\!\!& \jr & \+ & \jr & \je &\\[0.2em]
\petit4\!\!& \jr & \+ & \je &\\[0.2em]
\petit5\!\!& \jr & \je & &\\[0.2em]
\petit6\!\!& \je & & &\\
\end{array}
$
\caption{Aligned up is the top reduced pipe dream}
\end{subfigure}
\endgroup
\caption{Two reduced pipe dreams for [164235] obtained by aligning the diagram to the left and to the top.} \label{fig:topbottomrpd}
\end{figure}
The core of $1\pi'$ is the simplicial complex obtained by restricting $\PD(\pi)$ to the vertices corresponding to the positions of the crosses in the superimposition of these two pipe dreams. We refer to the region itself as the \emph{core region}, and denote it by $\rmcr(\pi)$. See Figure~\ref{fig:crpi} for an example. Note that different permutations can have the same core region, as is the case for $[15342]$ and $[15432]$.
\begin{figure}[h]
\begin{tikzpicture}
\draw (0,1.5)--(1.5,1.5)--(1.5,2.5)--(0.5,2.5)--(0.5,0)--(0,0)--(0,2)--(1.5,2);
\draw (0,0.5)--(1,0.5)--(1,2.5);
\draw (0,1)--(1,1);
\end{tikzpicture}\caption{The core region of $[164235]$}\label{fig:crpi}
\end{figure}
In~\cite{rc}, Bergeron and Billey introduced an algorithm to construct all reduced pipe dreams for $\pi$. Given a reduced pipe dream $P$ for all permutations $\pi$, a ladder admitting rectangle is a connected $k\times 2$ rectangle inside $P$ such that $k\geq2$ and the only $\textelbow$ inside this rectangle are in the top row and in the southeast corner, see the diagram on the left in Figure~\ref{fig:ladder}. A \emph{ladder move} on $P$ moves the $\textcross$ in the southwest corner of a ladder admitting rectangle to the northeast corner. Notice that the resulting pipe dream is a reduced pipe dream for $\pi$.
\begin{figure}[h]
\[
\begingroup
\arraycolsep=5pt
\begin{aligned}
\begin{array}{cc} {\color{white} \perm2{}}&{\color{white}\perm1{}}\\[0.5em]
\jr & \jr \\[0.2em]
\+ & \+ \\[0.2em]
\+ & \+\\[0.2em]
\+ & \+ \\[0.2em]
\+ & \je \\
\end{array}
\end{aligned}
\qquad\quad \lmp{} \qquad
\begin{aligned}
\begin{array}{cc} {\color{white} \perm2{}}&{\color{white}\perm1{}}\\[0.5em]
\jr & \+ \\[0.2em]
\+ & \+ \\[0.2em]
\+ & \+\\[0.2em]
\+ & \+ \\[0.2em]
\jr & \je \\
\end{array}
\end{aligned}
\endgroup
\]
\caption{Ladder move.}\label{fig:ladder}
\end{figure}
\begin{theo}[\cite{rc}]\label{thm:ladder}
The set of all reduced pipe dreams of $\pi$ equals the set of pipe dreams that can be derived from the bottom reduced pipe dream by a sequence of ladder moves.
\end{theo}
The \emph{boundary} of a pure simplicial complex $\Delta$ is the simplicial complex $\partial\Delta$ with facets the codimension~1 faces of $\Delta$ that are in exactly one facet of $\Delta$. A face $F$ of $\Delta$ is \emph{interior} if $F$ is not in $\partial\Delta$.
\section{Background on root polytopes}\label{sec:root}
We follow the exposition of~\cite[Section~4]{groth} in this section. A root polytope of type $A_{n}$ is the convex hull of the origin and some points in $\Phi^+=\{e_i-e_j \mid1\leq i0$, $(i,j)\in T(\pi)$. If $\mbox{ } T(\pi)=([m], \{(i,i+1) \mid i \in [m-1]\})$, then $e_1-e_m=\sum_{(i,j)\in T(\pi)} (e_i-e_j)$ is an interior point; moreover, $e_k-e_l$ for $1\leq k0$.
\end{proof}
\begin{theo}\label{thm:core}
Let $\pi=1\pi'$, with $\pi'$ dominant. Let $v$ be the unique vertex in $\C(\pi)$ on the ray between $0$ and $e_1-e_m$. For $\pi \neq 1 m (m-1)\dots 2$, $v$ is in the boundary of $\C(\pi)$, and $\core(\pi)$
is realized by the induced triangulation of the vertex figure of $\C(\pi)$ at $v$. For $\pi=1 m (m-1)\dots 2$, $\core(\pi)$ is realized by the induced triangulation of the boundary of $\C(\pi)$.
\end{theo}
\begin{proof} The vertex $v$, which is the unique vertex in $\C(\pi)$ on the ray between $0$ and $e_1-e_m$, is the unique coning point of the geometric realization of $\Scal(\pi)$. If $v$ is in the boundary of $\C(\pi)$, which happens exactly when $\pi \neq 1 m (m-1)\dots 2$ by Lemma~\ref{int}, then the induced triangulation of the vertex figure at $v$ of $\C(\pi)$ is a geometric realization of $\core(\pi)$ (which is homeomorphic to a ball). For $\pi=1 m (m-1)\dots 2$, the coning point $v$ lies in the interior of $\C(\pi)$, then since it is the only point in the interior of the canonical triangulation $\C(\pi)$ by Lemma~\ref{int}, then the induced triangulation of the boundary of $\C(\pi)$ is a geometric realization of $\core(\pi)$ (which is homeomorphic to a sphere).
\end{proof}
We now proceed to prove an auxiliary lemma used in the proof of Theorem~\ref{geom}.
\begin{lemm}\label{dim}
The core of the pipe dream complex of $\pi=1\pi'$, where the diagram of $\pi'$ is a partition, is of dimension $t(\pi)-2$. The dimension of the root polytope $\P(T(\pi))$ is $t(\pi)$ and its vertex figure at $0$ is of dimension $t(\pi)-1$. In particular, both $\Scal(\pi)$ and $\C(\pi)$ are of dimension $t(\pi)-1$.
\end{lemm}
\begin{proof} Since subword complexes are pure, then the dimension of the core of the pipe dream complex of $\pi$ equals the dimension of one of its facets. Consider the facet given by the bottom reduced pipe dream drawn inside the core region. The dimension of this facet equals one less than the number of elbows in the core and from the construction of $T(\pi)$ this equals $t(\pi)-2$. The dimension of the root polytope $\P(T(\pi))$ is the number of edges in $T(\pi)$, which by definition is $t(\pi)$.
\end{proof}
\subsection{Bijection between reduced pipe dreams for \texorpdfstring{$\pi$}{pi} and simplices of \texorpdfstring{$\C(\pi)$}{C(pi)}}
The map $M$ can be easily extended to a map between pipe dreams $P$ of $\pi$ drawn inside $R(\pi)$ and forests $F$ on $m$ vertices as follows. For each elbow tile in $P$ add the edge $(i,j)$ corresponding to the box of the elbow to $F$. Moreover, add the edge $(1,m)$ to $F$.
\begin{theo}\label{thm:bij}
The reduced pipe dreams of $\pi=1\pi'$, where the diagram of $\pi'$ is a partition, are in bijection with the noncrossing alternating spanning trees of the directed transitive closure of $T(\pi)$ via the map $M$.
\end{theo}
We prove Theorem~\ref{thm:bij} by induction on the number of columns in the diagram. We break it down in several lemmas.
\begin{lemm}\label{lem:base}
Take the permutation $\pi=1\pi'$, where the diagram of $\pi'$ is $\lambda=(k)$. The reduced pipe dreams of $\pi$ are in bijection with the noncrossing alternating spanning trees of the directed transitive closure of $T(\pi)$ via the map M.
\end{lemm}
\begin{proof}
The edges of $T(\pi)$ for such a $\pi$ are $(1,2)$ and $(2,j)$ for $j=3,\dots,k+2$ and thus for the transitive closure of $T(\pi)$ we add the edges $(1,j)$ with $j=3,\dots,k+2$. See Figure~\ref{fig:4}.
\begin{figure}[h]
\[
\begin{aligned}
\begin{tikzpicture}
\draw (0,0.5)--(0,0)--(0.5,0)--(0.5,2.5)--(1,2.5)--(1,0.5)--(0,0.5)--(0,2)--(1,2);
\draw (0,2)--(0,2.5)--(.5,2.5);
\draw (0,1.5)--(1,1.5);
\draw (0,1)--(1,1);
\draw (0.25,0.25) node {$\bullet$};
\draw (0.75,0.75) node {$\bullet$};
\draw (0.75,1.25) node {$\bullet$};
\draw (0.75,1.75) node {$\bullet$};
\draw (0.75,2.25) node {$\bullet$};
\draw (0.25,-0.25) node {1};
\draw (0.75,0.25) node {2};
\draw (1.25,0.75) node {3};
\draw (1.25,1.25) node {4};
\draw (1.25,1.75) node {5};
\draw (1.25,2.25) node {6};
\end{tikzpicture}
\end{aligned}
\quad \lmp{M}\quad
\begin{aligned}
\begin{tikzpicture}
\path (1,0) edge [bend left=60] (2,0);
\path (2,0) edge [bend left=60] (3,0);
\path (2,0) edge [bend left=60] (4,0);
\path (2,0) edge [bend left=60] (5,0);
\path (2,0) edge [bend left=60] (6,0);
\draw[] (1,-0.2) node {1};
\draw[] (2,-0.2) node {2};
\draw[] (3,-0.2) node {3};
\draw[] (4,-0.2) node {4};
\draw[] (5,-0.2) node {5};
\draw[] (6,-0.2) node {6};
\end{tikzpicture}
\end{aligned}
\]
\caption{The region $R(\pi)$ and the tree $T(\pi)$ for $\lambda=(4)$} \label{fig:4}
\end{figure}
For $l=2,\dots,k+2$ let $T_l$ be a tree on the vertex set $[k+2]$ consisting of the edges $(2,i)$ for $2