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%Notation for cluster variables (irreducible factors of domino entries)
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% Notation for BCFW coordinates
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%visible comments

\def\MSB#1{\textcolor{red}{[MSB: #1]}}
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\usepackage{lipsum}
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%%To delete
\def\MP#1{\textcolor{violet}{[MP: #1]}}


%% define your title in the usual way
\title[The magic number conjecture for the
$m=2$ amplituhedron and Parke--Taylor Polytopes]{ \texorpdfstring{The Magic Number Conjecture for the $m=2$ Amplituhedron
and Parke--Taylor Polytopes}{The Magic Number Conjecture for the m=2 Amplituhedron
and Parke--Taylor Polytopes}}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[M. Parisi, M. Sherman-Bennett, R. Tessler, L. Williams]{
\texorpdfstring{
Matteo Parisi\thanks{\href{mailto:mparisi@cmsa.fas.harvard.edu}{mparisi@cmsa.fas.harvard.edu}.}\addressmark{1}, Melissa Sherman-Bennett\thanks{Supported by NSF Award No. DMS-2103282}\addressmark{2}, Ran Tessler\addressmark{3}\thanks{Supported by ISF grants No. 335/19 and 1729/23},\\ \and Lauren Williams\thanks{Supported by NSF Award No. DMS-2152991}\addressmark{4}}{M. Parisi, M. Sherman-Bennett, R. Tessler, L. Williams}}


%% then use \addressmark to match authors to institutions here
\address{  
\addressmark{1} CMSA, Harvard University, Cambridge, MA;\\
\addressmark{2} Department of Mathematics, UC Davis, Davis, CA; \\ \addressmark{3} Department of Mathematics, Weizmann Institute of Science, Israel;\\ \addressmark{4} Department of Mathematics, Harvard University, Cambridge, MA.}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you do not have one yet
%% please do not use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please do not use \cite, and instead write out your citation as something like (author year)
\abstract{The \emph{amplituhedron} $\mathcal{A}_{n,k,m}$ is a geometric object which generalizes 
the positive Grassmannian (when $n=k+m$) and cyclic polytopes (when $k=1$). It was originally introduced in the context of \emph{scattering amplitudes}. Of substantial interest are the \emph{tilings} of the amplituhedron, which are analogous to triangulations of a polytope.
%In \cite{karp2020decompositions}, it was observed that the known tilings of 
%$\mathcal{A}_{n,k,2}$ have cardinality $\binom{n-2}{k}$ and the known tilings of 
%$\mathcal{A}_{n,k,4}$ have cardinality the \emph{Narayana number} $\frac{1}{n-3}\binom{n-3}{k+1}\binom{n-3}{k}$;
%generalizing these observations, 
In \cite{karp2020decompositions}, it was conjectured that for even $m$ the tilings of $\mathcal{A}_{n, k,m}$ have cardinality
the \emph{MacMahon number}, the number of plane partitions which fit inside a $k \times (n-k-m) \times \frac{m}{2}$ box.
We refer to this prediction as the \emph{Magic Number Conjecture}. 
In this paper we prove the Magic Number Conjecture for the $m=2$ amplituhedron:
that is, we show that 
each tiling of $\mathcal{A}_{n,k,2}$ has cardinality $\binom{n-2}{k}.$ 
We prove this by showing that all  positroid tilings of the hypersimplex
$\Delta_{k+1,n}$ have cardinality $\binom{n-2}{k},$ then applying
 \emph{T-duality}.  In addition, we give volume formulas for 
 \emph{Parke--Taylor polytopes} and tree positroid polytopes 
 in terms of circular extensions of \emph{cyclic partial orders}; and we prove
	new variants of the 
	classical \emph{Parke--Taylor identities}.}

%% put your French abstract here, or comment this out if you do not have one
% \resume{\lipsum[2]}

%% put your keywords here, or comment this out if you do not have them yet
\keywords{positroid, amplituhedron, hypersimplex, tile, tiling, MacMahon number}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{sample.bib}
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}

The (tree) \emph{amplituhedron} $\mathcal{A}_{n,k,m}(Z)$ is the image of the
positive Grassmannian $\Gr_{k,n}^{\scriptscriptstyle\geq 0}$ under the
\emph{amplituhedron map} $\tilde{Z}: \Gr_{k,n}^{\scriptscriptstyle\geq 0} \to \Gr_{k,k+m}$, a map induced by matrix multiplication by 
a positive matrix $Z\in \Mat_{n,k+m}^{>0}$.  The amplituhedron was introduced by
Arkani-Hamed and Trnka \cite{arkani-hamed_trnka} in order to give a
geometric  interpretation of
\emph{scattering amplitudes} in $\mathcal{N}=4$ super Yang--Mills theory. %;
%more specifically, they reformulated the BCFW recurrence for computing
%scattering amplitudes as giving a \emph{tiling} of the $m=4$ amplituhedron.
Central to this interpretation are \emph{tilings} of $\AA$, which are decompositions of $\AA$ into \emph{tiles} (see \Cref{def:tiling0}). 
%comes from a collection of $km$-dimensional cells
%of $\Gr_{k,n}^{\geq 0}$ on which $\tilde{Z}$ is injective, such that the images of the cells are disjoint and cover a dense subset of the amplituhedron.  
The notion of tiling can be thought of as a generalization of the notion of triangulation of a polytope.\footnote{We do not require tiles to intersect in a common ``face".}

While the case $m=4$ is most directly relevant to physics, 
the amplituhedron 
{\parfillskip = 0pt \par} \noindent
$\mathcal{A}_{n,k,m}(Z)$
makes sense for any positive $n,k,m$ such that $k+m \leq n$, and
has a very rich geometric and combinatorial
structure.  It  generalizes cyclic polytopes (when $k=1$),
cyclic hyperplane arrangements \cite{karpwilliams}
(when $m=1$), and the positive Grassmannian (when $k=n-m$), and it
is connected to
the hypersimplex and the positive tropical Grassmanian \cite{LPW,PSW} (when $m=2$). This paper will focus on the case $m=2$.  %In this case, the amplituhedron $\mathcal{A}_{n,k,2}$
%is also closely
%related to some scattering amplitudes, correlators of determinant operators and form factors in planar $\mathcal{N}=4$ super Yang--Mills \cite{Kojima:2020tjf,Caron-Huot:2023wdh,Basso:2023bwv}.

In \cite{karp2020decompositions} it was observed that the known tilings of
$\mathcal{A}_{n,k,2}(Z)$ have cardinality $\binom{n-2}{k}$,  the known tilings of
$\mathcal{A}_{n,k,4}(Z)$ have cardinality the Narayana number $\frac{1}{n-3}\binom{n-3}{k+1}\binom{n-3}{k}$,
and all tilings of 
$\mathcal{A}_{n,1,m}$ for even $m$
have cardinality $\binom{n-1- \frac{m}{2}}{\frac{m}{2}}$.\footnote{Every triangulation of the cyclic polytope
$C(n,m)$ contains exactly $\binom{n-1- \frac{m}{2}}{\frac{m}{2}}$ simplices when $m$ is even
\cite{bayer_93, rambau_97} and for $C(n,m)$ tilings and triangulations coincide \cite{OT10}.}
Based on these observations, \cite[Conjecture 8.1]{karp2020decompositions} conjectured that 
when $m$ is even, $\mathcal{A}_{n,k,m}$ has a tiling with cardinality
\begin{equation*}
M_{n,k,m}:=M\left(k, n-k-m, \frac{m}{2}\right), \ \ \text{ where } \ \ 
M(a,b,c):=\prod_{i=1}^a \prod_{j=1}^b \prod_{\ell=1}^c \frac{i+j+\ell-1}{i+j+\ell-2}
\end{equation*} 
is the \emph{MacMahon number}.
 This number has many remarkable interpretations, e.g. 
 {\parfillskip=0pt \par} \noindent
 $M(a,b,c)$ counts the number of \emph{plane partitions} which fit inside an $a \times b \times c$ box, collections of $c$ noncrossing lattice paths 
inside an $a\times b$ rectangle, perfect matchings of the honeycomb lattice, and more (see \cite{macmahon, bodroza_gutman_cyvin_tosic_1988}). 
%(Plane partitions also naturally appear in the computation of the Euler characteristic of the Hilbert scheme of points in a $3$-fold \cite{MR1382733}, and in the study of Donaldson--Thomas invariants \cite{maulik2006gromov,maulik2006gromov2}.)
\cite{Galashin_Lam_2020} later conjectured that \emph{each} tiling
should have this cardinality. 
We refer to the prediction that tilings of the amplituhedron $\mathcal{A}_{n,k,m}(Z)$ should have cardinality $M_{n,k,m}$ as the \emph{Magic Number Conjecture}. 

\textbf{Main results.}
The main result of this abstract is the proof of the Magic Number Conjecture for the $m=2$ amplituhedron (\Cref{cor:magic-number}).
% \comm{Recall that the \emph{hypersimplex} $\Delta_{k+1,n}$ is the convex
%hull of all $0/1$ vectors $e_I\in \R^n$, where $I$ is a $(k+1)$-element subset of $[n]$.  It is the \emph{uniform matroid polytope} 
%and it is also the image of the positive Grassmannian 
%$\Gr_{k+1,n}^{\geq 0}$ under the \emph{moment map}.}
We first prove that each tiling of the hypersimplex $\Delta_{k+1,n}$ by 
positroid polytopes 
has cardinality $\binom{n-2}{k}$ (\Cref{thm:magic-number}). In the special case tilings are finest regular subdivisions of $\Delta_{k+1,n}$ into positroid polytopes, this recovers \cite{SW21}. 
We also prove the more general result that 
if $\Gamma_{\mathcal{M}}$ is a full-dimensional positroid polytope,
all of its tilings have the same cardinality (\Cref{cor:positroidpolytopes}). 
The key ingredient for the proofs is to show that, although tiles do not have the same volume (see \Cref{fig:tiling}), there is an additive invariant on $\Delta_{k+1,n}$ defined in terms of \emph{Parke--Taylor functions} (\Cref{def:PT}) that is constant for all tiles (\Cref{prop:weight_tile}). 
%This invariant is defined in terms of \emph{Parke--Taylor functions} (\Cref{def:PT}) on $w$-simplices, which provide a triangulation of $\Delta_{k+1,n}$ and each tile.
Parke--Taylor functions have numerous connections e.g. with the cohomology of the moduli space $\mathcal{M}_{0,n}$ of $n$ points on the Riemann sphere in relation with \emph{scattering equations} \cite{Cachazo:2013hca} and \emph{Lie polynomials} \cite{Frost:2019fjn}.
Then we use \Cref{thm:magic-number} and apply the T-duality theorem from 
\cite[Theorem 11.6]{PSW} (which appears here as \Cref{thm:Tduality}) to prove the Magic Number Conjecture for the $m=2$ amplituhedron (\Cref{cor:magic-number}).
%\begin{mainthm}[Magic Number Theorem for $\Ank$]
%Suppose a collection $\mathcal{C}$ of cells of $\Gr_{k,n}^{\geq 0}$
%gives rise to a tiling of $\Ank$ for all $Z\in \Mat_{n,k+m}^{>0}$.
%Then $\mathcal{C}$ has cardinality $M_{n,k,2}=\binom{n-2}{k}$.
%\end{mainthm}
%\begin{mainthm}\label{mainthm:B}
%Every positroid tiling of the hypersimplex $\Delta_{k+1, n}$ consists of $\binom{n-2}{k}$ tiles. 
%\end{mainthm}
%We also establish some necessary conditions (\Cref{prop:covering-ngon-necessary}) for a collection of tiles to give rise to a tiling of $\Ank$ (equivalently, $\Delta_{k+1, n}$) in terms of the combinatorics of \emph{bicolored subdivisions} (cf. \Cref{def:tricolored}).
Finally, we introduce \emph{Parke--Taylor polytopes} (\Cref{def:ptpolytope}), whose volume equals the number of \emph{circular extensions} of a certain partial cyclic order (\Cref{prop:volume}). For each Parke--Taylor polytope we associate a combinatorial identity among Parke--Taylor functions involving such circular extensions (\Cref{thm:gen-parke-taylor}). 
% Our third main result is a formula for the volume of certain polytopes as the number of \emph{circular extensions} of a \emph{partial cyclic order}. 
 As corollary, we obtain a formula for the volume of the positroid polytopes which are tiles for $\Delta_{k+1,n}$ (\Cref{rk:tile_volume}). Complete arguments for these results appear in \cite{PSBTW}.
 %and a proof of the \emph{shuffle identities} for Parke--Taylor functions, whose special case include the \emph{$U(1)$-decoupling identity}, which is well-known in physics.
 
% The following result appears as \Cref{cor:vol-of-tile}; see \Cref{prop:volume} for a generalization.
% \begin{mainthm}
%Let $\sigma$ be a \emph{bicolored subdivision} (see \Cref{def:bicolored}), let $\Gamma_{\sigma}$ be the corresponding positroid polytope (see \Cref{thm:hypersimplex-tiles-ineq-facets}), and let $C_{\sigma}$ be the corresponding partial cyclic order (see \Cref{def:cyclic-from-perm-subdiv}).  Then the normalized volume of $\Gamma_{\sigma}$ equals the number of circular extensions of $C_{\sigma}.$
% \end{mainthm}
%  
%
%We note that \cite{AJR_cyclicOrders} also studied some polytopes contained in $[0,1]^n$ whose volumes can be 
%computed in terms of circular extensions of partial cyclic orders; they were motivated in part by 
%\cite[Exercise 4.56(d)]{EC1}.  The results of \cite{AJR_cyclicOrders} were subsequently generalized in \cite{Morales}.  In general our polytopes are different from the polytopes of \cite{AJR_cyclicOrders} although
%in a very special case they agree, see 
%\Cref{rem:AJVR}.
%
%A key tool in our proof of the Magic Number Theorem is the \emph{Parke--Taylor} function\footnote{From the seminal paper of the physicists Stephen J. Parke and T.R. Taylor \cite{PTs}.} associated to a permutation (see \Cref{def:PT}).

%\begin{definition}\label{def:PT}
%Let $w=w_1 \dots w_n \in S_n$ be a permutation on $[n]$. 
%The \emph{Parke--Taylor function} of $w$ is
%\begin{equation}
%    \PT(w):=\frac{1}{P_{w_1 \, w_2} P_{w_2 \, w_3} \ldots P_{w_{n} \, w_1}},
%\end{equation}
%where the $P_{ij}$ are Pl\"ucker coordinates on $\Gr_{2,n}$, with the convention that Pl\"ucker coordinates are antisymmetric in their indices.
%\end{definition}
%
%
%For example, if $w=2143$ then 
%$$\PT(w) = \frac{1}{P_{21} P_{14} P_{43} P_{32}} = 
%-\frac{1}{P_{12} P_{14} P_{34} P_{23}}.$$
%
%We view the Parke--Taylor function $\PT(w)$ 
%as a rational function on (the affine cone over) the Grassmannian $\Gr_{2,n}$.  It is well-defined
%on (the cone over) the top-dimensional matroid stratum of $\Gr_{2,n}$, that is, the locus 
%where no Pl\"ucker coordinates vanish. 


 

%As a corollary of our work, we obtain many relations for Parke--Taylor functions (cf. \Cref{thm:gen-parke-taylor}), one for each \emph{tricolored subdivision} of an $n$-gon (cf. \Cref{def:tricolored}), in terms of circular extensions of certain cyclic partial orders. Moreover, we prove the known \emph{shuffle identities} for Parke--Taylor functions as below.

%\begin{proposition*}
%Let $I\subseteq [n-1]$ and fix a permutation $u=u_1 \dots u_r$ of $I$ and a permutation $v=v_1 \dots v_{n-r-1}$ of $[n-1]\setminus I$. Let $\Sh_n(u,v)$ be the set of permutations in $S_n$ in which $u_1,\ldots,u_r$ appear in order and $v_1,\ldots,v_{n-r-1}$ appear in order, and which send $n$ to $n$. Then we have
%\begin{equation*}
% \sum_{w\in \Sh_n(u,v)} \PT(w) = 0.   
%\end{equation*}
%\end{proposition*}


%The result above appears later as \Cref{prop:shufflePT}.    
%In the special case that $I=\{1\}$,
%this result recovers the ``classical'' \emph{$U(1)$-decoupling identity} for Parke--Taylor functions (cf. \Cref{cor:U1}).
% In \Cref{thm:gen-parke-taylor} we give a generalization of \Cref{thm2:gen-parke-taylor}, where the sum is over circular extensions of certain cyclic partial orders. 



%\bigskip
%
%The structure of this paper is as follows. In \Cref{sec:background} we provide
%background on the positive Grassmannian, the hypersimplex, and the amplituhedron, including a well-known triangulation of the hypersimplex into $w$-simplices. In \Cref{sec:combtiles} we give a simple combinatorial characterization of exactly which $w$-simplices lie in a given tile for the hypersimplex. In \Cref{sec:cyclic}, we introduce partial cyclic orders and use them to rephrase the results of the previous section. In \Cref{sec:PT}, we define the weight function of a positroid polytope using Parke--Taylor functions; prove that the weight function of each tile $\Gamma_{\sigma}$ of $\Delta_{k+1,n}$ is constant; and deduce the Magic Number Theorem.  In \Cref{sec:combtiling}, we extend the notion of weight function to facets of tiles and prove additional necessary conditions for when a collection of bicolored subdivsions gives rise to a positroid tiling of $\Delta_{k+1,n}.$  In \Cref{sec:combPT}, we introduce Parke--Taylor polytopes and prove new relations among Parke--Taylor functions.   
%Finally in \Cref{sec:G} we give an interpretation of our previous results in terms of the $G$-amplituhedron.
%\bigskip




\section{Background}

%We review background on the positive Grassmannian, the hypersimplex, and the amplituhedron. We introduce the notion of positroid tiles and tilings of a space $X$ which is the surjective image of the positive Grassmannian. We review the characterization of positroid tiles for the hypersimplex \cite{LPW} and for the $m=2$ amplituhedron \cite{PSW} using the combinatorics of bicolored subdivisions; we also state the T-duality theorem of \cite{PSW} which connects positroid tilings of the hypersimplex and $m=2$ amplituhedron. Finally, we discuss a triangulation of the hypersimplex into $w$-simplices \cite{StanleyTriangulation,SturmfelsGrobner,LamPost}, which is the simultenous refinement of all positroid tilings and is essential to the proofs in later sections.

%\textbf{The (positive) Grassmannian.}
The \emph{Grassmannian} $\Gr_{k,n} = \Gr_{k,n}(\mathbb{R})$
is the space of all $k$-dimensional subspaces of the $\mathbb{R}^n$ vector space.
Let $[n]$ denote $\{1,\dots,n\}$, and $\binom{[n]}{k}$ denote the set of all $k$-element 
subsets of $[n]$.
We can
 represent a point $V \in
\Gr_{k,n}$  as the row-span 
of
a full-rank $k\times n$ matrix $C$ with entries in
$\mathbb{R}$.  
Then for $I=\{i_1 < \dots < i_k\} \in \binom{[n]}{k}$, the \emph{Pl\"ucker coordinate} $P_I(C)$ is the $k\times k$ minor of $C$ using the columns $I$. 
The {Pl\"{u}cker coordinates} of $V$ are independent of the choice of matrix
representative $C$ (up to common rescaling). The \emph{Pl\"ucker embedding} 
$V \mapsto \{P_I(C)\}_{I\in \binom{[n]}{k}}$
embeds  $\Gr_{k,n}$ into
projective space\footnote{
We will sometimes abuse notation and identify $C$ with its row-span.}. 

\begin{definition}[{\cite{lusztig, postnikov}}]\label{def:positroid}
We say that $V\in \Gr_{k,n}$ is \emph{totally nonnegative}
     if (up to a global change of sign)
       $P_I(C) \geq 0$ for all $I \in \binom{[n]}{k}$.
Similarly, $V$ is \emph{totally positive} if $P_I(C) >0$ for all $I
      \in \binom{[n]}{k}$.
We let $\Grk$ and $\Gr_{k,n}^{>0}$ denote the set of
totally nonnegative and totally positive elements of $\Gr_{k,n}$, respectively.  
The set $\Grk$ is called the \emph{totally nonnegative}  \emph{Grassmannian}, or
       sometimes just the \emph{positive Grassmannian}.
\end{definition}

If we partition $\Grk$ into strata based on which Pl\"ucker coordinates are strictly
positive and which are $0$, we obtain a cell decomposition of $\Grk$
into \emph{positroid cells} \cite{postnikov} that glue together to form a CW complex \cite{PSWv1}.
Each positroid cell $S$ gives rise to a matroid $\mathcal{M}$, whose bases are precisely
the $k$-element subsets $I$ such that the Pl\"ucker coordinate
$P_I$ does not vanish on $S$; the matroid $\mathcal{M}$ 
is called a \emph{positroid}.

We will be interested in certain images of positroid cells.

%\noindent \textbf{Positroid tiles and tilings.}
%\comm{Given any surjective map $\phi$ from a cell complex onto a topological space $X$ of dimension $d$, it is natural to try to decompose $X$ using images
%of $d$-dimensional cells under $\phi$.  
%We now consider $X$ to be the positive Grassmannian $\Grk$.}
% This leads to the definition of \emph{$\phi$-induced tiling} \cite[Definition 2.19]{WilliamsICM} \MP{also our paper?} which we apply
%in the case that our cell complex is the positive Grassmannian.\footnote{There are many reasonable
%variations of this definition.  One might want to relax the injectivity
%assumption, or to impose further restrictions on how boundaries of the 
%images of cells should overlap. Note that in the literature, positroid tilings are sometimes
%called \emph{(positroid) triangulations}. We avoid this terminology in order to avoid confusion with the notion of e.g. polytopal triangulations.}  
%\comm{When $\phi$ is the amplituhedron map $\tilde{Z}$,
%tilings were first studied in 
%\cite{arkani-hamed_trnka} (where they were referred to as \emph{triangulations}).}
\begin{definition}\label{def:tiling0} 
	Let $\phi: \Grk \to X$ be 
 a continuous surjective map where $\dim X=d$. The closure $\overline{\phi(S)}$ of the image of a positroid cell $S \subset \Grk$ is a \emph{positroid tile} for $X$ if $\phi$ is injective on $S$ and $\dim S =d$.  
A \emph{positroid tiling} of $X$ is a collection $\{\overline{\phi(S)}\}_{S \in \mathcal{C}}$ of positroid tiles that cover $X$ and have pair-wise disjoint interiors.
%\begin{itemize}
%	\item (disjointness): open tiles $\phi(S)$
%		and $\phi(S')$ for distinct positroid cells $S$ and $S'$ in $\mathcal{C}$ are disjoint
%              \item (covering): $\cup_{S \in \mathcal{C}}\overline{\phi(S)} = X$.
%	\end{itemize}
\end{definition}

We will focus here on positroid tilings for two different maps $\phi$: the moment map $\mu$ and the amplituhedron map $\tZ$, which maps $\Grk$ onto the hypersimplex and the amplituhedron, respectively. %We will discuss the combinatorics of tiles and tilings for these choices of $\phi$, which turn out to be closely related.

\noindent\textbf{The hypersimplex.}
Let $\{e_1, \dots, e_n\}$ denote the standard basis of $\R^n$, and define $e_I := \sum_{i \in I} e_i \in \R^n$. If $x \in \R^n$ and $I \subset [n]$, we set $x_I:=\sum_{i \in I} x_i$. For $i \neq j \in [n]$, the cyclic interval from $i$ to $j$ is $[i,j]:=\{i, i+1, \dots, j-1, j\}$. 

\begin{definition}
The \emph{$(k+1,n)$-hypersimplex} 
$\Delta_{k+1,n} \subset \R^n$
 is the convex hull of the points $e_I$ where $I$ runs over $\binom{[n]}{k+1}$. 
\end{definition}


%The \emph{moment map} from $\Gr_{k+1,n}$ to $\R^n$
%is defined as follows: \MP{need def of moment map?}
%\begin{definition}[The moment map] \label{def:mom_map}
%	Let $A$ be a $(k+1) \times n$ matrix representing a point of 
%	$\Gr_{k+1,n}$.
%	The \emph{moment map} 
%	$\mu: \Gr_{k+1,n} \to \R^n$ is defined by 
%	$$\mu(A) = \frac{ \sum_{I \in \binom{[n]}{k+1}} |p_I(A)|^2 e_I}
%	{\sum_{I \in \binom{[n]}{k+1}} |p_I(A)|^2}.$$
%\end{definition}

The hypersimplex $\Delta_{k+1,n}$ is the image of the Grassmannian $\Gr_{k+1,n}$ (or the positive Grassmannian $\Gr_{k+1,n}^{\geq 0}$, \cite[Proposition 7.10]{tsukerman_williams}) under the \emph{moment
map}. 
%If one restricts the moment map to $\Gr_{k+1,n}^{\geq 0}$ then the image is again the hypersimplex
%$\Delta_{k+1,n}$ 
%\cite[Proposition 7.10]{tsukerman_williams}.
The hypersimplex $\Delta_{k+1,n}$ is $(n-1)$-dimensional, as it is contained in the hyperplane $x_{[n]}=k+1$. %See \Cref{fig:HYS24} for an example of the hypersimplex.

\begin{remark}\label{rem:hypercube}
The projected hypersimplex
$\pi(\Delta_{k+1,n})$ under $\pi \!:\!(x_1,\dots,x_n) \!\mapsto\! (x_1,\dots,x_{n-1})$ is the slice of the
unit hypercube $\mbox{\mancube}_{n-1}$ contained between $x_{[n-1]} = k$ and $x_{[n-1]}= k+1$.
%we see that the hypersimplex is linearly equivalent to the polytope in
%$\R^{n-1}$ defined by
%$$\{(x_1,\dots,x_{n-1}) \ \vert \
%0 \leq x_i \leq 1; k \leq x_{[n-1]} \leq k+1\}.$$
%That is, the projected hypersimplex
%$\pi(\Delta_{k+1,n})$ is the slice of the
%unit hypercube $\mbox{\mancube}_{n-1}$ contained between the hyperplanes
%$x_{[n-1]} = k$ and $x_{[n-1]}= k+1$.
\end{remark} 


The moment map images of positroid cells can easily be described using positroids:

\begin{proposition}[{\cite[Proposition 7.10]{tsukerman_williams}}]
    Let $S \subset \Gr_{k+1, n}^{\ge 0}$ be a positroid cell, $\mathcal{M}$ its corresponding positroid, and $\Gamma_{\mathcal{M}} := \mbox{conv}\{e_I : I \in \mathcal{M}\}$ its associated \emph{positroid polytope}. Then $\overline{\mu(S)}=\htile{\mathcal{M}}$.
\end{proposition}



Specializing \Cref{def:tiling0}, with $\phi$ being the moment map $\mu$ and $X$ being the hypersimplex $\Delta_{k+1,n}$, we denote the positroid tiles for $\Delta_{k+1,n}$ as  $\Gamma_{S}:=\overline{\mu(S)}$.
%\begin{definition} Let $\mu: \Gr^{\geq 0}_{k+1,n} \to \Delta_{k+1,n}$ be the moment map.
% The positroid polytope $\htile{\mathcal{M}} =\overline{\mu(S)}$, which is the closure of the image of a positroid cell $S$, is a \emph{tile} for $\Delta_{k+1,n}$ if $\mu$ is injective on $S$ and $\dim S=n-1$.
% A \emph{positroid tiling} of $\Delta_{k+1,n}$ is a collection of tiles with disjoint interiors whose union is $\Delta_{k+1, n}$.
%\end{definition}



\noindent\textbf{The amplituhedron.}
Building on \cite{abcgpt,hodges},
Arkani-Hamed and Trnka
\cite{arkani-hamed_trnka}
defined
the \emph{(tree) amplituhedron} as 
the image of the positive Grassmannian under a positive linear map.
Let $\Mat_{n,p}^{>0}$ denote the set of $n\times p$ matrices whose maximal minors
are positive.

\begin{definition}\label{definition_amplituhedron}
Let $Z\in \Mat_{n,k+m}^{>0}$, where $k+m \leq n$.
    The \emph{amplituhedron map}
$\tilde{Z}:\Gr_{k,n}^{\ge 0} \to \Gr_{k,k+m}$
        is defined by
        $\tilde{Z}(C):=CZ$,
    where
 $C$ is a $k \times n$ matrix representing an element of
        $\Gr_{k,n}^{\ge 0}$,  and  $CZ$ is a $k \times (k+m)$ matrix representing an
         element of $\Gr_{k,k+m}$.
        The \emph{amplituhedron} $\mathcal{A}_{n,k,m}(Z) \subset \Gr_{k,k+m}$ is the image
$\tilde{Z}(\Gr_{k,n}^{\ge 0})$.
\end{definition}

%It is convenient to use the following coordinates when discussing points in $\AA$. 
%\begin{definition}[Twistor coordinates]\label{def:tw_coords}
%Fix
%  $Z \in \Mat_{n,k+m}^{>0}$
%   with rows  $Z_1,\dots, Z_n \in \R^{k+m}$.
%   Given  $Y \in \Gr_{k,k+m}$
%   with rows $y_1,\dots,y_k$,
%   and $\{i_1,\dots, i_m\} \subset [n]$,
%   we define the \emph{twistor coordinate} $
%	\llrr{{Yi_1} {i_2} \cdots {i_m}}$
%        to be  the determinant of the
%        matrix with rows
%        $y_1, \dots,y_k, Z_{i_1}, \dots, Z_{i_m}$.
%\end{definition}
%\noindent Note that the twistor coordinates are defined only up to a common scalar multiple.

Specializing \Cref{def:tiling0}, with $\phi$ being the amplituhedron map $\tilde{Z}$ and $X$ being the amplituhedron $\AA$, we denote the positroid tiles for $\AA$ as  $\atile{S}:=\overline{\tilde{Z}(S)}$.

%\begin{definition} Let $\tilde{Z}: \Gr^{\geq 0}_{k,n} \to \Gr_{k,k+m}$ be the amplituhedron map.
% The image $\gto{S} :={\tilde{Z}(S)}$ of a positroid cell $S$ is an \emph{open tile} for $\AA$ if $\tilde{Z}$ is injective on $S$ and $\dim S =mk$. The closure $\atile{S} :=\overline{\tilde{Z}(S)}$ of an open tile is a \emph{tile} for $\AA$.
% A \emph{positroid tiling} of $\AA$ is a collection $\{\atile{S}\}_{S \in \mathcal{C}}$ of tiles whose union is $\AA$ and such that the open tiles $\{\gto{S}\}_{S \in \mathcal{C}}$ are pairwise disjoint.
%\end{definition}


In this work it is convenient to work with ``all-$Z$'' tilings, as defined below.

\begin{definition}\label{def:tiling}
We call $\{Z_{S} \}_{S\in \mathcal{C}}$ an \emph{all-$Z$ tiling} of the amplituhedron $\mathcal{A}_{n,k,m}$ if 
		$\{Z_{S} \}_{S\in \mathcal{C}}$ is a tiling of $\mathcal{A}_{n,k,m}(Z)$
  for all  $Z\in \Mat_{n,k+m}^{>0}$.
\end{definition}

%\begin{remark}
%We will drop the phrase ``all-$Z$'' as we will always consider all-$Z$ tilings\footnote{Currently, there is no known example of tiling which is not an all-Z tiling.}.
%\end{remark}

%There is a natural notion of  \emph{facet} of a tile, 
%generalizing the notion of facet of a polytope. 
%
%\begin{definition}[Facet of a cell and a tile]\label{def:facet2} 
%Given two positroid cells $S'$ and $S$, we say that 
%$S'$ is a \emph{facet} of $S$ if 
%$S' \subset \partial{S}$ and $S'$ has codimension $1$ in $\overline{S}$.
%If $S'$ is a facet of $S$ and $Z_S$ is a tile of $\AA$, we say that $Z_{S'}$ is a
%	\emph{facet} 
%of $Z_{S}$ if 
%      $\gt{S'} \subset \partial \gt{S}$ and has codimension 1 in $\gt{S}$.
%\end{definition}


\noindent\textbf{Correspondence between the hypersimplex and the amplituhedron.} 
\begin{definition}\label{def:tricolored}  
Let  $\mathbf{P}_n$ be a convex $n$-gon with vertices labeled from
$1$ to $n$ in clockwise order.
       A \emph{tricolored subdivision $\tdiv$} is a partition of 
       $\mathbf{P}_n$ into black, white, and grey polygons such that two polygons sharing an edge have different colors.  
       We say that $\tdiv$ has 
       \emph{type $(k,\ell, n)$} if any triangulation of the black (respectively, grey) polygons consists of exactly $k$ black (respectively, $\ell$ grey) triangles.

A \emph{bicolored subdivision} $\sigma$ of type $(k,n)$ is a tricolored subdivision of type $(k,0,n)$, that is, with no grey polygons. See \Cref{fig:tricolored}.
\end{definition}

%\begin{remark}
% Bicolored subdivisions of type $(k,n)$ are in bijection with \emph{separable permutations} on $[n-1]$ with $k$ descents \cite[Corollary 12.6]{PSW}, which are enumerated by a refinement of the \emph{large Schr\"oder numbers} \cite[A175124]{OEIS}.   
%\end{remark}


\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
 \includegraphics[width=0.42\textwidth]{tricolor.pdf}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=0.42\textwidth]{bicoloredsubs.pdf}
\end{subfigure}
 \caption{(Left): A tricolored subdivision $\tdiv$ of type $(3,2,8)$.  It gives rise to the cyclic order
        $C_{\tdiv}$ which is the union of the chains
        $C_{(2,5,7)}$, $C_{(5,7,6)}$, and $C_{(1,8,7,2)}$. (Right): A bicolored subdivision $\sigma$ of type $(5,9)$, it corresponds to a tile $\Gamma_{\sigma}$ in $\Delta_{6,9}$.} 
\label{fig:tricolored}
\end{figure}




In \cite{PSW}, a subset of the authors of this work showed that positroid tiles for both $\Delta_{k+1,n}$ and $\mathcal{A}_{n,k,2}(Z)$ are in bijection with the bicolored subdivisions of type $(k,n)$\footnote{these are enumerated by a refinement of the \emph{large Schr\"oder numbers} \cite[A175124]{OEIS} and are in bijection with \emph{separable permutations} on $[n-1]$ with $k$ descents \cite[Corollary 12.6]{PSW}.}. From a bicolored subdivision $\sigma$ of type $(k,n)$ one can directly read the inequalities describing the associated positroid tile $\Gamma_\sigma$ for $\Delta_{k+1,n}$ and the positroid tile $Z_\sigma$ for $\mathcal{A}_{n,k,2}(Z)$.
In particular, tiles of the hypersimplex 
$\Delta_{k+1,n}$ and tiles of the amplituhedron
$\mathcal{A}_{n,k,2}$ are in bijection with each other, and this bijection induces a bijection on tilings as well:
\begin{theorem}[{\cite[Theorem 11.6]{PSW}}]\label{thm:Tduality}
Let $\mathcal{S}$ be a collection of bicolored subdivisions of type $(k,n)$. Then
$\{\htile{\sdiv} \}_{\sdiv\in \mathcal{S}}$ is a positroid tiling of $\Delta_{k+1,n}$ if and only if $\{\atile{\sdiv} \}_{\sdiv\in \mathcal{S}}$ is a tiling of $\A_{n, k, 2}$.    
\end{theorem}


%In \cite[Proposition 9.5, 9.6]{PSW} a subset of the authors of the present work 
%showed that positroid tiles for 
%$\mathcal{A}_{n,k,2}(Z)$ are in bijection with the bicolored subdivisions of type
%$(k,n),$
%
%
%The following characterization of tiles of $\Delta_{k+1,n}$ in terms of bicolored subdivisions of type $(k,n)$ appears in \cite{PSW} as Proposition 9.5 and 9.6;
%
%In \cite[Theorem 4.25]{PSW}, a subset of the authors of the present work 
%showed that positroid tiles for 
%$\mathcal{A}_{n,k,2}(Z)$ are in bijection with the bicolored subdivisions of type
%$(k,n),$ by associating a plabic graph to each bicolored subdivision and showing 
%that these plabic graphs exactly correspond to the $2k$-dimensional cells on 
%which the amplituhedron map is injective.  Moreover, \cite[Theorem 4.28]{PSW}
%gives a concise description of these positroid tiles in terms of inequalities.
%%These two results are summarized in the following theorem.
%%
%%
%%\begin{theorem}\label{thm:amp-tiles-facets-inequalities}
%%The positroid tiles for $\mathcal{A}_{n,k,2}(Z)$ are in bijection with the bicolored subdivisions of type $(k,n).$  Moreover, if $\sigma$ is such a bicolored subdivision, then the corresponding open tile is
%%$$Z^{\circ}_\sigma = \{Y\in \Gr_{k,k+2} \ \vert \ 
%% (-1)^{\area(i\to j)}\llrr{Y i j} >0 \text{ for all compatible arcs }i\to j \text{ of }\sigma \text{ with }i<j\}.$$
%%Alternatively, let $\tilde{\sigma}$ be a triangulation of the black polygons of $\sigma$. Then 
%%\begin{enumerate}
%%    \item $Z^{\circ}_{\sigma}$ is cut out by the inequalities $(-1)^{\area(i\to j)}\llrr{Y i j} >0$ for all arcs $i\to j$ in $\tilde{\sigma}$ with $i<j$. 
%%    \item the facets of $Z_\sigma$ lie on the hypersurfaces $\llrr{Y i j} = 0$, where $i\to j$ a facet-defining arc of $\sigma$. 
%%\end{enumerate}
%%    \end{theorem}
%%\begin{example}
%%    For $\sigma$ in \Cref{fig:unpunctured}: if $Y \in Z^{\circ}_{\sigma}$ then
%%    $(-1)^{2}\llrr{Y 3 7} >0$ ($3 \rightarrow 7$ is a compatible arc).
%%    There is a facet of $Z_{\sigma}$ lying on $\llrr{Y 6 4}=0$ ($6 \rightarrow 4$ is a facet-defining arc).
%%\end{example}
%
%
%
%From \Cref{thm:hypersimplex-tiles-ineq-facets,thm:amp-tiles-facets-inequalities}, we see that tiles of $\Delta_{k+1, n}$ and $\Ank$ are both in bijection with bicolored subdivisions of type $(k,n)$, and thus 

%There is a particularly nice class of tilings which we can now describe in terms of 
%bicolored subdivisions.
%
%\begin{definition}[Bicolored subdivisions of kermit type]\label{def:kermit}
%Let $v\in [n]$, let $I = \{i_1,\dots,i_k\}\in \binom{[n]\setminus \{v\}}{k}$ and let 
%$\sigma_I$ be the bicolored subdivision of the $n$-gon obtained by drawing 
%black triangles with vertices $\{v, i_\ell, i_{\ell}+1\}$ for $\ell=1,\dots,k$. 
%(We then repeatedly merge black triangles sharing a common edge into a larger
%black polygon.)  We say that $\sigma^v_I$ is a \emph{kermit bicolored subdivision based at vertex $v$}. We will often denote $\sigma^1_I$ by $\sigma_I$.
%\end{definition}
%
%\begin{figure}[ht]
%\includegraphics[width=0.8\textwidth]{kermit.pdf}
%        \caption{The bicolored subdivisions (based at vertex $v=1$) giving rise to the kermit tilings of 
%        $\Delta_{3,6}$ and of $\mathcal{A}_{6,2,2}$.}
%        \label{fig:tilings}
%\end{figure}
%
%
%
%\begin{proposition}\cite[Proposition 11.32]{PSW}\label{prop:tilings}
%Choose $0 \leq k \leq n-2$ and let $I$ run over $\binom{[2,n-1]}{k}$.  Then the collections
%$\{\Gamma_{\sigma_I}\}$ and $\{Z_{\sigma_I}\}$ are positroid tilings of 
%$\Delta_{k+1,n}$ and $\mathcal{A}_{n,k,2},$ respectively.  In particular,
%$$\Delta_{k+1,n} = \bigcup_I \Gamma_{\sigma_I} \text{ and }
%\mathcal{A}_{n,k,2}(Z) = \bigcup_I Z_{\sigma_I}.$$
%Moreover if we apply a cyclic shift to our definitions
%(i.e. rotate the labels of the $n$-gon), we obtain
%positroid tilings of $\Delta_{k+1,n}$ and $\mathcal{A}_{n,k,2}$.
%\end{proposition}
%
%For any $v\in [n]$, we refer to the tiling $\{\Gamma_{\sigma_I^v}\}$ %as the \emph{descent} tiling
%of $\Delta_{k+1,n}$, and the tiling $\{Z_{\sigma_I^v}\}$ %as the \emph{sign-flip} tiling 
%of $\mathcal{A}_{n,k,2},$ as \emph{kermit tilings (based at $v$)}, see \Cref{fig:tilings}. 


See \cite[Table 1]{PSW} for many other correspondences between the hypersimplex and the $m=2$ amplituhedron.

\noindent\textbf{The triangulation of the hypersimplex into $w$-simplices.} 
%We now review a unimodular triangulation of the hypersimplex, originally due to Stanley \cite{StanleyTriangulation}, and also studied by Sturmfels \cite{SturmfelsGrobner} and Lam--Postnikov \cite{LamPost}. The maximal simplices of this triangulation are indexed by permutations with a fixed number of descents, which are counted by the Eulerian numbers. We follow the presentation of \cite{LamPost}. 


\begin{definition} 
Let $w \in S_n$. A letter $i<n$ is a \emph{left descent}\footnote{Left descents, sometimes called \emph{recoils} in the literature, are discussed extensively 
in \cite[Chapter 1]{bjornerbrenti}.} of $w$ if $i$ occurs to the right of $i+1$ in $w$, i.e. 
 $w^{-1}(i) > w^{-1}(i+1)$.
We say $i\in [n]$ is a \emph{cyclic left descent} of $w$ if either
 $i<n$ is a left descent of $w$ or if $i=n$ and $1$ occurs to the left of $n$ in $w$.
We let $\cdes(w)$ be the set of cyclic left descents of $w$. 
 We often omit the word `left' in `cyclic left descent'.
\end{definition}

%\comm{\begin{remark}
%Left and right descents and descent sets are discussed extensively 
%in \cite[Chapter 1]{bjornerbrenti}.
%Left descents are sometimes called \emph{recoils} in the literature.
%\end{remark}}

\begin{definition}\label{Dk+1n}
Choose $0 \leq k \leq n-2$. We let $D_n$ be the set of permutations $w \in S_n$ with $w_n=n$ and $D_{k+1, n}$ to be the set of permutations $w \in D_n$ with $k+1$ cyclic descents. 
\end{definition}
Note that $|D_{k+1, n}|$ equals the \emph{Eulerian number} $E_{k,n-1}:=\sum_{\ell=0}^{k+1}(-1)^{\ell} \binom{n}{\ell}(k+1-\ell)^{n-1}$. 

\begin{example}\label{ex:D24}
  The set $D_{2,4}$ contains the following $E_{2,3}=4$ permutations.
  
  \begin{center}
      \begin{tabular}{|c|c|c|c|c|}
        \hline  $w$ & 1324 & 3124 & 2134 & 2314 \\
        \hline  $\cdes(w)$ & $\{2,4\}$ & $\{2,4\}$ & $\{1,4\}$ & $\{1,4\}$ \\ \hline
      \end{tabular}
  \end{center}
  %$w_1=1324, w_2=3124, w_3=2134, w_4=2314$. Moreover, we have: $\cdes(w_1)=\{2,4\}$, $\cdes(w_2)=\{2,4\}$, $\cdes(w_3)=\{1,4\}$ and $\cdes(w_4)=\{1,4\}$.
\end{example}

%It will often be convenient for us to identify elements of $D_n$ with $n$-cycles in $S_n$. In \Cref{sec:cyclic}, we will also identify elements of $D_n$ and $n$-cycles with total cyclic orders.

 For $w=w_1 w_2 \dots w_n \in S_n$, we write $(w)=(w_1 \ w_2 \cdots w_n)$ for the cycle $w_1 \mapsto w_2 \mapsto \cdots \mapsto w_n \mapsto w_1$. The map $w \mapsto (w)$ bijects $D_n$ onto the set of $n$-cycles in $S_n$.



\begin{definition}[$w$-simplices] \label{definition:wsimplexHSimplex}
 For $w=w_1 w_2 \dots w_n \in D_{k+1, n}$, let $w^{(a)}$ denote the cyclic rotation of $w$ ending at $a$. We denote\footnote{Note $I_r$  depends only on $(w)$ rather than $w$ itself.} $I_r=I_r(w):=\cdes(w^{(r)})$ and define the \emph{w-simplex} $\simp{w} \subseteq \Delta_{k+1, n}$ to be the convex hull of $e_{I_1}, \dots, e_{I_n}$; this is an $(n-1)$-dimensional simplex. 
% We call $I_{w_1} \to I_{w_2} \to \dots \to I_{w_n} \to I_{w_1}$ the \emph{circuit} of $\simp{w}$ and sometimes abuse notation by equating $\simp{w}$ and its circuit.
\end{definition}

\begin{example}\label{ex:w_simplex}
    By \Cref{ex:D24}, $\Delta_{2,4}$ has four $w$-simplices, see \Cref{fig:HYSwsimp}. The vertices of $\simp{w}$ with $w= 1324$ are computed from the cyclic descents of the rotations of $w$.

\begin{center}
    \begin{tabular}{|c|c|c|c|c|}\hline
     $r$ & 1 & 2 & 3 & 4 \\ \hline
       $w^{(r)}$ & 3241 & 4132 & 2413 & 1324 \\ \hline
      $I_r=\cdes(w^{(r)})$ & $\{1,2\}$  & $\{2,3\}$ & $\{1,3\}$ & \{2,4\} \\ \hline
    \end{tabular}
\end{center}
    

 Then $\Delta_{(1324)}$ is the convex hull of  $e_{12},e_{23},e_{13},e_{24}$ (the top-right red simplex in \Cref{fig:HYSwsimp}). 

%\MP{Put the 2 figures of hypersimplices next to each other to save space.}
%    \begin{figure}[ht]
%\includegraphics[width=0.22\textwidth]{HYSwsimp.pdf}
%	\caption{$w$-simplices for $\Delta_{2,4}$. We omit parentheses in subscripts for readability.}
%	\label{fig:HYSwsimp}
%\end{figure}
\end{example}



\begin{figure}[ht]
\centering
\begin{subfigure}{.5\textwidth}
  \centering
 \includegraphics[width=0.4\textwidth]{HYS24.pdf}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=0.4\textwidth]{HYSwsimp.pdf}
\end{subfigure}
 \caption{(Left): The hypersimplex $\Delta_{2,4}$ projected in $\mathbb{R}^3$. (Right): $w$-simplices for $\Delta_{2,4}$. We omit parentheses in subscripts for readability.}
	\label{fig:HYSwsimp}
\end{figure}

%\begin{remark}\label{rmk:convention-change}
%Our notation differs compared to some previous works, see \cite[Remark 2.26]{} \MP{this paper} for details.
%
%%\cite[Section 10]{PSW} uses left cyclic descent tops rather than left cyclic descent bottoms. It also uses the notation $\Delta_w$ for $w$-simplices. The bijection $w \mapsto \simp{w}$ from $D_{k+1, n}$ to $w$-simplices in \Cref{definition:wsimplexHSimplex} is slightly different from the bijection $u \mapsto \Delta_{u}$ of \cite[Definition 10.3]{PSW}. In particular, let $u$ be the permutation obtained from $w$ by subtracting $1$ from each number and rotating so $n$ is at the end. Then $\simp{w}=\Delta_{u}$. The left cyclic descent tops of $u^{(r-1)}$ are the same as the left cyclic descent bottoms of $w^{(r)}$, so the sets $I_r(w)$ defined here are the same as the sets $I_r(u)$ in \cite[Definition 10.3]{PSW}.
%\end{remark}

%\begin{remark}\label{remark:circuitdefinition}
%For a $w$-simplex
%$\simp{w}$,
%the consecutive elements $I_{w_{i-1}} \to I_{w_{i}}$ of the circuit are related by 
%$$I_{w_{i}} = I_{w_{i-1}} \setminus\{w_i -1\} \cup \{w_i\}. $$
%(In particular, $r$ is always in $I_r$ and $r-1$ is never in $I_r$.)
%From this, one can see that, letting $u$ be as in \Cref{rmk:convention-change},  $\simp{w}$ agrees with the simplex denoted $\Delta_{(u)}$ in \cite[Section 2.4]{LamPost}. In particular, the circuit $I_{w_1} \to I_{w_2} \to \dots \to I_{w_n} \to I_{w_1}$ is the circuit used to define $\Delta_{(u)}$. 
%\end{remark}



The following triangulation of the hypersimplex first appeared in \cite{StanleyTriangulation},
though the description in terms of $w$-simplices appeared first in \cite{LamPost}.
\begin{proposition}[$\Delta_{k+1,n}$ is the union of $w$-simplices \cite{StanleyTriangulation}]\label{prop:StanleyTriangulation}
	The $w$-simplices $\{\simp{w}: w \in D_{k+1, n}\}$ are the maximal simplices of a triangulation of the 
 hypersimplex $\Delta_{k+1,n}$.
Moreover,
projecting $\{\simp{w}: w\in S_n\}$ into $\mathbb{R}^{n-1}$ (see \Cref{rem:hypercube}),
we obtain the maximal simplices in a triangulation of the hypercube $\mbox{\mancube}_{n-1}$ 
which refines the subdivision of the hypercube into hypersimplices. 
\end{proposition}

%%It follows from \cite[Theorem 2.7]{LamPost} that the triangulation
%%$$\Delta_{k+1, n}= \bigcup_{w \in D_{k+1, n}} \simp{w}$$
%is obtained by cutting the hypersimplex with all facet hyperplanes of all positroid tiles.
The triangulation into $w$-simplices is the simultaneous refinement of all positroid tilings \cite[Theorem 2.7]{LamPost}. It follows that:

\begin{proposition}[\cite{LamPost}] \label{prop:w-simp-triangulate-pos-poly}
 Every positroid tile\footnote{In fact, this statement hold for any full-dimensional positroid polytope.} for $\Delta_{k+1, n}$ has a triangulation into $w$-simplices.
\end{proposition}


%We will later need to discuss facet-sharing of $w$-simplices. The characterization of facet-sharing was given in \cite[Theorem 2.9]{LamPost}. The hyperplane on the shared facet lies can be deduced readily from \cite{LamPost} (we also prove it in \Cref{cor:ineq-and-facet-w-simp}).
%
%\begin{proposition}\label{prop:w-simp-facet-sharing}
%    Let $u,w \in D_{k+1, n}$. Then $\simp{w}$ and $\simp{u}$ share a facet if and only if $(w)$ and $(u)$ are related by swapping an adjacent pair of numbers, i.e. if $(w)=(A\ i \ j \ B)$ and $(u)=(A \ j \ i \ B)$, where $j \neq i \pm 1$. The shared facet, which we denote by $\simp{w}^{(ij)}$ or $\simp{u}^{(ji)}$, lies on a hyperplane $x_{[i,j-1]}=c$.
%\end{proposition}

%Finally, the results from this section have analogues in the $m=2$ amplituhedron.

%\MP{Not sure whether to mention the following:}
%\begin{remark}\label{rmk:w-chambers}
%    Cutting $\Delta_{k+1, n}$ with the facet hyperplanes of all positroid tiles decomposes the hypersimplex into $w$-simplices $\simp{w}$. Analogously, cutting $\Ank$ with the facet hypersurfaces of all positroid tiles decomposes the amplituhedron into \emph{$w$-chambers} ${\cham{w}}$ 
%    (cf. \Cref{def:ampchamber}), which are also indexed by $w \in D_{k+1, n}$ \cite[Corollary 11.20]{PSW}. Each chamber consists of points whose twistor coordinates have specified signs, which can be read off of $w$. Some $w$-chambers may be empty for a particular choice of $Z$, but all are nonempty for some $Z \in \Mat^{>0}_{n, k+2}$. 
%\end{remark}

\begin{remark}\label{rmk:wsimp-in-tile-iff-wcham-in-tile}
    The bijection between tiles of the hypersimplex and the amplituhedron can be further refined using $w$-simplices and \emph{$w$-chambers}. Each tile $\atile{\sdiv}$ of the amplituhedron is a union of $w$-chambers \cite[Corollary 10.17]{PSW}. If $\cham{w}$ is nonempty, then $\cham{w} \subset \atile{\sdiv}$ if and only if $\simp{w} \subset \htile{\sdiv}$ \cite[Proposition 11.1]{PSW}.
\end{remark}

\section{Results}

\textbf{The Magic Number Theorem for the $m=2$ Amplituhedron.}\label{sec:PT}
We show that every tiling of the hypersimplex $\Delta_{k+1, n}$ and every all-$Z$ tiling of $\mathcal{A}_{n,k,2}$ consists of $\binom{n-2}{k}$ tiles. 
%A key tool in our proof of the Magic Number Theorem is the \emph{Parke--Taylor} function\footnote{From the seminal paper of the physicists Stephen J. Parke and T.R. Taylor \cite{PTs}.} associated to a permutation.

\begin{definition}\label{def:PT} The \emph{Parke--Taylor function} of a permutation\footnote{$\PT(w)$ depends only on the $n$-cycle $(w)$; that is, if $(u)=(w)$, then $\PT(w)=\PT(u)$} $w=w_1 \dots w_n \in S_n$ is:
\begin{equation}
    \PT(w):=\frac{1}{P_{w_1 \, w_2} P_{w_2 \, w_3} \ldots P_{w_{n} \, w_1}},
\end{equation}
viewed as a rational function on $\widehat{\Gr}_{2,n}$, 
  the 
 affine cone over the Grassmannian $\Gr_{2,n}$, where the $P_{ij}$ are Pl\"ucker coordinates\footnote{with the convention that Pl\"ucker coordinates are antisymmetric in their indices} on $\Gr_{2,n}$. 
\end{definition}
%We remark that $\PT(w)$ depends only on the $n$-cycle $(w)$; that is, if $(u)=(w)$, then $\PT(w)=\PT(u)$.

%\comm{For example, if $w=2143$ then 
%$$\PT(w) = \frac{1}{P_{21} P_{14} P_{43} P_{32}} = 
%-\frac{1}{P_{12} P_{14} P_{34} P_{23}}.$$
%Parke--Taylor functions have numerous connections e.g. with the cohomology of the moduli space $\mathcal{M}_{0,n}$ of $n$ points on the Riemann sphere in relation with \emph{scattering equations} \cite{Cachazo:2013hca} and \emph{Lie polynomials} \cite{Frost:2019fjn}.} Let $\mathbf{I}_n= 1 \dots n$ denote the identity permutation on $[n]$.
%
%\MP{Put these two definitions together.}
\begin{definition}\label{def:wtPT}
Let the \emph{weight function} of a $w$-simplex for $\Delta_{k+1,n}$ be $\Omega(\simp{w}):=\PT(w)$.  Let $\Gamma \subset \Delta_{k+1, n}$ be a full-dimensional positroid polytope,
    or any other subset of $\Delta_{k+1,n}$ which is a union of $w$-simplices  (cf. 
    \Cref{prop:w-simp-triangulate-pos-poly}).  We define the \emph{weight function} of $\Gamma$ to be the sum of the weight functions of $w$-simplices included in $\Gamma$:
    \begin{equation*}
\Omega(\Gamma):=\sum_{\simp{w} \subset \Gamma} \Omega(\simp{w}).
    \end{equation*}
\end{definition}

The main step to prove the magic number conjecture for $\mathcal{A}_{n,k,2}$ is the following: 

\begin{proposition}\label{prop:weight_tile}
  Let $\htile{\sdiv}$ be a tile of $\Delta_{k+1,n}$. Then the weight function of $\htile{\sdiv}$ is
  \begin{equation}\label{eq:weight_tile}
      \Omega(\htile{\sdiv})=(-1)^k \PT(\mathbf{I}_n),
  \end{equation}
  with $\mathbf{I}_n \in S_n$ the identity permutation. Hence, all tiles of $\Delta_{k+1,n}$ have the same weight function.
\end{proposition}
% \MP{Sketch of proof, we use some residue arguments and induction. We prove that $\Res_{P_{ij}=0}\Omega(\htile{\sdiv})=0$, for all $i,j$ such that $(i,j)$ is not an edge of a white or black polygon of $\sigma$ and  $\Res_{P_{i, i-1}=0}\Omega(\htile{\sdiv})=(-1)^c \Omega(\htile{\sdiv'})$, with $\sigma'$ obtained from $\sigma$ by contracting the boundary edge $(i-1,i)$. Plus homogeneity arguments.}

In contrast, the normalized volume of tiles is far from constant (see \Cref{fig:tiling}).

\begin{proposition}\label{prop:samesize}
Let $R \subset \Delta_{k+1,n}$ be a subset of $\Delta_{k+1,n}$ which admits a positroid tiling, i.e. it can be written 
as the union
of positroid tiles $\{\htile{\sdiv}\}_{\sdiv \in \mathcal{S}}$ whose interiors are disjoint.  Then all positroid tilings of $R$ have the same cardinality.    
\end{proposition}
\begin{proof}
For \emph{any} tiling $\{\htile{\sdiv}\}_{\sdiv \in \mathcal{S}}$ of $R$, we have
    \begin{equation*}
        \Omega(R)=\sum_{\sdiv \in \mathcal{S}} \sum_{\simp{w} \subset \htile{\sdiv}}\Omega(\simp{w})=\sum_{\sdiv \in \mathcal{S}} \Omega(\htile{\sdiv})=|\mathcal{S}| (-1)^k \PT(\mathbf{I}_n),
    \end{equation*}
    where for the first and second equality we used that $\htile{\sdiv}=\cup_{\simp{w} \subset \htile{\sdiv}} \simp{w}$ and that the tiles $\{\htile{\sdiv}\}_{\sdiv \in \mathcal{S}}$ have disjoint interiors and cover $\Delta_{k+1,n}$. For the last equality we used \Cref{prop:weight_tile}.
        It follows that each positroid tiling of $R$ must have the same cardinality.
    \end{proof}

By \Cref{prop:samesize}, and the fact that $\Delta_{k+1, n}$ admits tilings of size $\binom{n-2}{k}$ \cite{BaoHe}, we deduce:

\begin{theorem}\label{thm:magic-number}
Every positroid tiling of $\Delta_{k+1, n}$ consists of $\binom{n-2}{k}$ tiles.
\end{theorem}


%\begin{example}
%Consider the collection $\mathcal{S}$ of bicolored subdivisions in \Cref{fig:tiling}. Then $\{\htile{\sdiv}\}_{\sdiv \in \mathcal{S}}$ is a tiling for $\Delta_{4,7}$. This can be verified by checking that each $w$-simplex in $\Delta_{4,7}$ lies in a unique tile $\htile{\sdiv}$ with $\sdiv \in \mathcal{S}$. By T-duality (cf. \Cref{thm:Tduality}), $\{\atile{\sdiv}\}_{\sdiv \in \mathcal{S}}$ is an all-Z tiling of $\mathcal{A}_{7,3,2}$. The tiling contains $10=\binom{7-2}{3}=M_{7,3,2}$ tiles (cf. \Cref{cor:magic-number}).
%\end{example}


\begin{figure}[ht]
\center
\includegraphics[width=0.7\textwidth]{ex_tiling.pdf}
        \caption{A collection of $10=\binom{7-2}{3}=M_{7,3,2}$ bicolored subdivisions of type $(3,7)$ (labelled from \textcircled{\raisebox{-0.9pt}{1}} to \textcircled{\raisebox{-0.9pt}{\scriptsize 10}}) which gives a tiling for $\Delta_{4,7}$ and $\mathcal{A}_{7,3,2}$. The number in the box below each bicolored subdivision $\sigma$ is the volume of the corresponding positroid polytope $\Gamma_\sigma$ in $\Delta_{4,7}$, which equals $\mbox{Ext}(C_\sigma)$. E.g. for \textcircled{\raisebox{-0.9pt}{5}} we have: $C_\sigma=C_{(1247)}\cup C_{(243)}\cup C_{(4765)}$ and $\mbox{Ext}(C_\sigma)=26$. The sum of their volumes is the Eulerian number $E_{3,6}=302$.}
        \label{fig:tiling}
\end{figure}

 Using results on the positive Dressian, one can show that every full-dimensional positroid polytope has a positroid tiling; see \cite[Proposition 5.14]{PSBTW}. Therefore, by \Cref{prop:samesize}:

\begin{corollary}\label{cor:positroidpolytopes}
All positroid tilings of a full-dimensional positroid polytope $\Gamma_{\mathcal{M}} \subset \Delta_{k+1,n}$ have the same cardinality.
\end{corollary}

It would be interesting to find an explicit combinatorial formula for such cardinality. 

Using \Cref{thm:Tduality} and \Cref{thm:magic-number} we can deduce the Magic Number Conjecture:  
\begin{theorem}\label{cor:magic-number}
 Every all-$Z$ positroid tiling of $\mathcal{A}_{n,k,2}$ consists of $M_{n,k,2}= \binom{n-2}{k}$ tiles.   
\end{theorem}

We can also get the amplituhedron-analogue of \Cref{prop:samesize}:

\begin{proposition}\label{prop:ampanalogue}
If $R$ is a full-dimensional subset of $\mathcal{A}_{n,k,2}(Z)$ which admits an all-$Z$ positroid tiling, then every all-$Z$ positroid tiling of $R$ has the same cardinality.  
\end{proposition}

An interesting case for \Cref{prop:ampanalogue} is when $R$ is (the closure of) the full-dimension\-al 
image of a positroid cell $S$ under the amplituhedron map $\tilde{Z}$, also called \emph{Grasstope}.


%Another combinatorial necessary condition for a collection of tiles to give  a tiling is:
%
%\begin{proposition}\label{prop:covering-ngon-necessary}
%Suppose that a collection $\mathcal{S}$ of bicolored subdivisions indexes a positroid tiling of $\Delta_{k+1, n}$ (or an 
%all-$Z$ tiling of $\mathcal{A}_{n,k,2}$).
%    Then we can partition the multi-set of black polygons in $\mathcal{S}$ into $\binom{n-3}{k-1}$ subdivisions of the $n$-gon, see \Cref{fig:overlap_tilings}.
%\end{proposition}
%
%\begin{figure}[ht]
%\includegraphics[width=0.9\textwidth]{ex_covering.pdf}
%\caption{Partitioning the multi-set of black polygons in $\mathcal{S}$ of \Cref{fig:tiling}  into $\binom{7-3}{3-1}=6$ subdivisions of the $7$-gon, each made of polygons with the same color or letter.}
%        \label{fig:overlap_tilings}
%\end{figure}

\noindent \textbf{Circular extensions: Parke--Taylor polytopes and identities.}
\emph{Circular extensions} and \emph{partial cyclic orders} are analogues of linear extensions and partial orders.
\begin{definition}\label{def:cyclic-order}
    A \emph{(partial) cyclic order} on a finite set $X$ is a ternary relation $C \subset X^3$ such that for all $a, b, c, d \in X$:
    \begin{align}
        (a,b,c) \in C &\implies (c, a, b) \in C \tag{cyclicity}\\
        (a,b,c) \in C &\implies (a,c, b) \notin C \tag{asymmetry}\\
        (a,b,c) \in C \text{ and }(a, c, d) \in C &\implies (a,b,d) \in C \tag{transitivity}
    \end{align}
    A cyclic order $C$ is \emph{total} if for all $a, b, c\in X$, either $(a,b,c)\in C$ or $(a, c, b) \in C$.
  \end{definition}

 Informally, a total cyclic order $C$ on $[n]$ is a way of placing $1, \dots, n$ on a circle, just as a total order is a way of placing $1, \dots, n$ on a line. Given $w=w_1\dots w_n \in S_n$, we denote as $C_w$ the total cyclic order obtained by placing $w_1, w_2, \dots, w_n$ on the circle clockwise. We also identify $C_w$ with the $n$-cycle $(w)$. 
% \comm{Note that each total cyclic order on $[n]$ is of the form $C_w$ for a unique permutation $w \in D_n$ (cf. \Cref{Dk+1n} for the definition of $D_n$). We move interchangeably between $w \in D_n$, the $n$-cycle $(w)$ and the total cyclic order $C_w$.}


\begin{definition}\label{def:circular-extension}
    A total cyclic order $C$ is a \emph{circular extension} of a cyclic order $C'$ if $C' \subset C$. We let $\Ext(C)$ denote the set of all\footnote{Not all cyclic orders have a circular extension \cite{Meg_cyclicOrders}, that is, 
 $\Ext(C)$ could be empty.} circular extensions of a cyclic order $C$. 
    % In an abuse of notation, if $C$ is a partial cyclic order on $[n]$, we sometimes write $(w) \in \Ext(C)$ if $C_w \in \Ext(C)$.
\end{definition}

%\comm{Not all cyclic orders have a circular extension \cite{Meg_cyclicOrders}, that is, 
% $\Ext(C)$ could be empty.  Moreover, the problem of determining
%whether a cyclic order has a circular extension is NP-complete
%\cite{GalilMegiddo}.}

\begin{definition}
Let $x_1,\dots,x_m$ be a sequence of $m$ distinct elements of $[n]$.
The \emph{chain} $C_{(x_1, x_2 \dots, x_m)}$ is the cyclic order in which $x_1, x_2 \dots, x_m$ appear on a circle ordered clockwise.
%for each triple $1\leq i < j < \ell \leq m$ 
%        we have $(x_i,x_j, x_{\ell}) \in C$ (which implies by cyclicity that
%        also $(x_j,x_{\ell}, x_i)$ and $(x_{\ell}, x_i, x_j)$ lie in $C$).  
%We call this partial cyclic order a \emph{chain}.
\end{definition}

We now associate a partial cyclic order to every tricolored subdivision. See \Cref{fig:tricolored} for an example.
 
%\MP{Make an example referring to this picture that will go into the background.}
%\begin{figure}[ht]
%\center
%\includegraphics[height=1.4in]{tricolor.pdf}
%        \caption{A tricolored subdivision $\tdiv$ of type $(3,2,8)$.  It gives rise to the cyclic order
%        $C_{\tdiv}$ which is the union of the chains
%        $C_{(2,5,7)}$, $C_{(5,7,6)}$, and $C_{(1,8,7,2)}$.}
%        \label{fig:tricolored}
%\end{figure}







\begin{definition}[$\tdiv$-order]\label{def:cyclic-from-perm-subdiv}
Let $\tdiv$ be a tricolored subdivision of $\mathbf{P}_n$ which includes $q$ non-grey polygons $P_1, \dots, P_q$.
If $P_a$ is white (respectively, black), we let $v_1,\dots,v_r$ denote its list of vertices read in clockwise (respectively, counterclockwise) order.  We then associate the chain $C_a = C_{(v_1,\dots,v_r)}$ to $P_a$.  Finally, we define the \emph{$\tdiv$-order} $C_{\tdiv}$
to be the cyclic order which is the union of the cyclic orders associated
to the black and white polygons $C_1,\dots, C_q$, see \Cref{fig:tricolored}.
\end{definition}
%One can easily verify that $C_{\tdiv}$ is a cyclic order. See \Cref{fig:tricolored} for an example.
 
% \MP{Result 1. Call it proposition?}
% \begin{corollary}\label{cor:w-simp-in-tile-cyclic-order-version}
% Let $\sdiv$ be a bicolored subdivision of type $(k,n)$. Then
% \[\htile{\sdiv}= \bigcup_{(w) \in \Ext(C_\sdiv)} \simp{w} \qquad \text{and} \qquad \atile{\sdiv}= \bigcup_{(w) \in \Ext(C_\sdiv)} \cham{w}.\]
% That is, $\htile{\sdiv}$ (resp. $\atile{\sdiv}$) is the union of $w$-simplices $\simp{w}$ (resp. $w$-chambers $\cham{w}$) where the $w$-order $C_w$ is a circular extension of the $\sdiv$-order $C_{\sdiv}$.
% \end{corollary}

% Because the $w$-simplices are unimodular, \Cref{cor:w-simp-in-tile-cyclic-order-version} gives rise to an expression for the normalized volume of each hypersimplex tile.

% \begin{corollary}\label{cor:vol-of-tile}
% Let $\sdiv$ be a bicolored subdivision.
%     The normalized volume of $\htile{\sdiv}$ is the number of circular extensions of $C_\sdiv$. That is, $\Vol(\htile{\sdiv})= |\Ext(C_\sdiv)|$.
% \end{corollary}

For each tricolored subdivision, we now introduce a polytope in $\mathbb{R}^{n-1}$.
We will need to work with the projection $\pi: \R^n \to \R^{n-1}$ from \Cref{rem:hypercube}.


\begin{definition}
 Given a bicolored subdivision $\sigma$, let 
$\ptile{\sigma}$ denote the projected polytope $\pi(\htile{\sigma})$.
Given a $w$-simplex $\Delta_{(w)}$, let $\psimp_{(w)}$ denote
the projected simplex $\pi(\Delta_{(w)})$.
\end{definition}


\begin{definition}
Let $\tdiv$ be a tricolored subdivision.
      Given a pair of vertices $i,j$ of $\mathbf{P}_n$, we say that 
       the arc $i \to j$ is \emph{compatible} with $\tdiv$ if the arc either 
       is an edge of a black, white, or grey polygon, 
       or lies  entirely inside a black or white polygon of $\tdiv$.  
If $i \to j$ is compatible with $\tdiv$, the \emph{area to the left of }$i \to j$ (respectively, \emph{grey area to the left of }$i \to j$), denoted by $\area(i \to j)$ (respectively, $\garea(i \to j)$), is the number of black triangles (respectively, grey triangles) to the left of $i \to j$ in any triangulation of the black (respectively, grey) polygons of $\tdiv.$  
\end{definition}

\begin{definition}\label{def:ptpolytope}
Let $\tdiv$ be a tricolored subdivision of $\mathbf{P}_n$ of type 
$(k,\ell,n).$
We define the 
\emph{Parke--Taylor}
polytope $\ptile{\tdiv} \subset \R^{n-1}$ by the following inequalities: 
    for any compatible arc $i \to j$ with $i<j, $
    $$ \area(i\to j) \leq x_{[i,j-1]}\leq \area(i\to j)+\garea(i\to j)+1.$$
\end{definition}

\begin{example}\label{ex:cube}
    If $\tdiv$ is the tricolored subdivision of $\mathbf{P}_n$ which is just a grey polygon on $n$ vertices, then the Parke--Taylor polytope
    $\ptile{\tdiv}$ is the unit hypercube $\mbox{\mancube}_{n-1} \subset \R^{n-1}$.  
\end{example}
%\MP{Say that when there is no grey polygon, we recover bicolored subdivisions and PT polytope is a positroid polyope which is a tile for bla.}

%\begin{example}
%    For the tricolored subdivision $\tdiv$ from \Cref{fig:tricolored} of type $(3,2,8)$, the Parke--Taylor  polytope $\ptile{\tdiv} \subset \mathbb{R}^7$ is defined by the inequalities $0 \leq x_i \leq 1$ for $1 \leq i\leq 7$, and 
%    \begin{align*}
%      3 \leq x_{[1,7]} \leq 6, \qquad 0 \leq x_{[2,4]} \leq 2, \qquad 1 \leq x_{[2,6]} \leq 4\\
%     1 \leq x_{[5,6]} \leq 2,  \qquad 2 \leq x_{[1,6]} \leq 5, \qquad 2 \leq x_{[2,7]} \leq 5.
%    \end{align*}
%%\comm{     Note that this is not a minimal description of $\ptile{\tdiv}$, as the inequality $1 \leq x_{[2,6]} \leq 4$ is implied by 
%%$1 \leq x_{[5,6]} \leq 2$ and $0 \leq x_{[2,4]} \leq 2$. }  
% For $\sigma$ in \Cref{fig:unpunctured}, if $x \in \Gamma_{\sigma}$ then $2 \leq x_{[3,6]} \leq 3$ ($3 \rightarrow 7$ is a compatible arc).
%    Some facet inequalities of $\Gamma_{\sigma}$ are: $x_5 \geq 0$ ($5$ is a vertex of a white polygon of $\sigma$), $x_{[6,3]}\geq 5$ ($6 \rightarrow 4$ is a facet-defining arc).
%\end{example}

\begin{remark}\label{rk:pt_pos_pol}
  In the case when $\tau$ has no grey polygons $(\ell=0)$, i.e. $\tau$ is a bicolored subdivision of type $(k,n)$, Parke--Taylor polytopes are (the projected) tree positroid polytopes, which are the tiles for $\Delta_{k+1,n}$. This follows from \Cref{def:ptpolytope} and \cite[Proposition 9.5]{PSW}.  
\end{remark}

Parke--Taylor polytopes admit positroid tilings, for example via the \emph{kermit subdivisions} (see \cite[Definition 2.23]{PSBTW})). Each $\ptile{\tdiv}$ can also be triangulated into projected 
$w$-simplices using the combinatorics of circular extensions of the cyclic $\tau$-order:

%\begin{figure}
%    \centering
%    \includegraphics[width=\textwidth]{PTpolytope_ex.pdf}
%    \caption{An illustration of the second part of \Cref{prop:tri-sub-polytope-as-union}. The polytope $\ptile{\sdiv}$ whose $\sdiv$ is on the left hand side and in \Cref{fig:tricolored} is the union of the polytopes $\ptile{\tau}$ for $\tau$ on the right hand side. The distinguished vertex of the grey polygon is $2$ in this case.}
%    \label{fig:pt-polytope-subdivision}
%\end{figure}

\begin{proposition}\label{prop:volume}
Let $\tdiv$ be a tricolored subdivision of type $(k,\ell,n)$. Then
\[\ptile{\tdiv}= \bigcup_{(w) \in \Ext(C_\tdiv)} \psimp_{(w)}.\]
It follows that the volume of $\ptile{\tdiv}$ is the number of circular extensions
of the cyclic order $C_{\tdiv}$. That is, $\Vol(\tilde{\Gamma}_{\tau})= |\Ext(C_\tdiv)|$, see \Cref{fig:tiling}.
\end{proposition}

%\begin{example} \MP{Change example.}
%    For the tricolored subdivision $\tdiv$ in \Cref{fig:tricolored}, we have 
%    \[\Vol(\ptile{\tdiv})=55+62+127+127=371\]
%    which is the number of circular extensions of the cyclic order 
%    $C_\tdiv=C_{(2,5,7)} \cup C_{(5,7,6)} \cup C_{(1,8,7,2)}.$
%    Each term in the sum above is the volume of $\ptile{\sdiv}$ for $\sdiv$ on the right hand side of \Cref{fig:pt-polytope-subdivision}.
%\end{example}

\begin{remark}\label{rk:tile_volume}   By \Cref{rk:pt_pos_pol},  \Cref{prop:volume} describes exactly which $w$-simplices triangulate the positroid polytope $\Gamma_\sigma$, which is a tile for $\Delta_{k+1,n}$. Moreover, it provides a formula for the volume of $\Gamma_\sigma$ as the number of circular extensions of the cyclic order $C_\sigma$.
\end{remark}

\begin{remark}
Part of \Cref{prop:volume} bears some analogies with a result from \cite{AJR_cyclicOrders}. The two results agree in a very special case, but otherwise deal with different contexts.
\end{remark}

%\MSB{I commented out from here}
For each tricolored subdivision, using \Cref{prop:weight_tile}, we also prove an identity among Parke--Taylor functions (cf. \Cref{def:PT}). 
%Some Parke--Taylor identities have previously been studied and connected to: non-planar generalizations of \emph{plabic graphs} \cite{Arkani-Hamed:2014bca, Cachazo:2018wvl}; the \emph{momentum amplituhedron} \cite{Damgaard:2021qbi}; \emph{Lie polynomials} \cite{Frost:2019fjn}; and \emph{CGEM amplitudes} in relation to the configuration of $n$ points in $\mathbb{CP}^{k-1}$ \cite{Cachazo:2023ltw}. 

\begin{theorem}[Parke--Taylor identities from tricolored subdivisions]\label{thm:gen-parke-taylor}
Let $\tdiv$ be a tricolored subdivision of $\mathbf{P}_n$ which contains at least one grey
polygon, and let $C_{\tdiv}$ be the corresponding
cyclic partial order.  Recall that 
$\Ext(C_{\tdiv})$ is the set of cyclic extensions of $C_{\tdiv}$.
 Then we have that 
$$\sum_{(w)\in \Ext(C_{\tdiv})} \PT(w) = 0.$$
\end{theorem}
% This can be interpreted as the `weight function' of Parke--Taylor polytopes is zero.
By specializing to particular tricolored subdivisions, we can use \Cref{thm:gen-parke-taylor} to prove some well-known Parke--Taylor identities from physics \cite{Frost:2019fjn}, such as the \emph{shuffle} and \emph{$U(1)$-decoupling identities} \cite[Proposition 7.17, Corollary 7.16]{PSBTW}.
%\MSB{To here}




%\begin{remark}
%Comparing with \Cref{prop:volume}, \Cref{thm:gen-parke-taylor} can be interpreted as the statement that the weight function of a Parke--Taylor polytope is zero, with the weight function of each $\tilde{\Delta}_{(w)}$ being $\PT(w)$. The proof uses tilings of Parke--Taylor polytopes into positroid polytopes and \Cref{prop:weight_tile}.
%\end{remark}

%If we consider particular tricolored subdivisions we recover some well-known Parke--Taylor identities from physics, such as



%\begin{figure}[ht]
%\includegraphics[height=1.1in]{pt_ids.pdf}
%        \caption{Tricolored subdivisions generating the relation in \Cref{thm2:gen-parke-taylor} (left) and $U(1)$ decoupling identities as a special case (right).}
%        \label{fig:pt_ids}
%\end{figure}

% \begin{remark}\label{rmk:Sn-action}
%     There is an $S_n$ action on $\Gr_{2,n}$, where the permutation $\pi$ acts by sending the matrix $A$ with columns $v_1, \dots, v_n$ to the matrix $\pi(A)$ with columns $v_{\pi(1)}, \dots, v_{\pi(n)}.$ We have that $\PT(w)$ evaluated on $\pi(A)$ is equal to $\PT(\pi w)$ evaluated on $A$. Thus, given any identity of Parke--Taylor functions $\PT(w)$, we can obtain another identity by left-multiplying each argument $w$ by a fixed permutation in $S_n$.
% \end{remark}

%We highlight a few special cases of \Cref{thm:gen-parke-taylor}. 
%

%\MSB{I also commented out this.}
%\begin{corollary}\label{thm2:gen-parke-taylor}
%Let $S=\{s_1< \dots < s_r\} \subsetneq [n-1]$, and let 
%$\mathcal{D}_S$ be the set of permutations in $D_n$ in which 
%$s_1,\dots, s_r$ appear in order.  Then we have that 
%$$\sum_{w\in \mathcal{D}_S} \PT(w) = 0.$$
%\end{corollary}
%
%In the specific case of \Cref{thm2:gen-parke-taylor} when $r=n-1$ and $S=[n-1]\setminus \{1\}$, we obtain the known \emph{$U(1)$ decoupling identities}\footnote{This relation has the interpretation of a scattering amplitude of $n-1$ $SU(N)$ `gluons' and a $U(1)$ `photon' that vanishes because the latter is decoupled from (i.e. does not interact with) the former.} for Parke--Taylor functions.
%\begin{corollary}\label{cor:U1} We have
%\begin{equation*}
%\PT(1,2,3,\ldots,n)+\PT(2,1,3,\ldots,n)+\PT(2,3,1,\ldots,n)+\ldots+\PT(2,3,\ldots,1,n)=0. 
%\end{equation*}
    
%\end{corollary}

%We end this section with a variation of \Cref{thm2:gen-parke-taylor}, which gives the \emph{shuffle identities} for Parke--Taylor functions (see \cite[Section 3.1]{Frost:2019fjn} and \cite{Cresson}). 
%\begin{proposition} \label{prop:shufflePT}
%Let $I\subseteq [n-1]$ and fix a permutation $u=u_1 \dots u_r$ of $I$ and a permutation $v=v_1 \dots v_{n-r-1}$ of $[n-1]\setminus I$. Let $\Sh_n(u,v)$ be the set of permutations in $D_n$ in which $u_1,\ldots,u_r$ appear in order and $v_1,\ldots,v_{n-r-1}$ appear in order. Then we have
%\begin{equation*}
% \sum_{w\in \Sh_n(u,v)} \PT(w) = 0.   
%\end{equation*}
%\end{proposition}


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