
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Equivalence of volume recursions}
\label{sec:D2VR}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we introduce another ``averaged volume'' recursion that 
interpolates between the $D_2$-recursion introduced in Section~\ref{sec:D2rec}
and the volume recursion in  Theorem~\ref{intro:VolRec}. We will show that 
the averaged volume recursion and the $D_2$-recursion give the same generating 
functions, and then Theorem~\ref{intro:IntFormula} will follow from it. 
\par
Recall from~\eqref{eq:Hindform} the definition of $\cHH_{\{i,j\}}\in R[[z_i, z_j]]$ for $i\neq j$, where 
$R=\QQ[h_1,h_2,\ldots]$. For any list of positive integers $\bfp=(p_1,\ldots, p_k)$ we define 
$$
\cHH_{\{i,j\}}^\bfp \= \frac{\partial^{k}}{\partial h_{p_1} \cdots 
\partial h_{p_k}} \cHH_{\{i,j\}}\,.
$$  
For a finite set  $I=\{i_1,\ldots,i_n\}$ of positive integers 
we define the formal series $\cAA_I \in R[[z_{i_1},z_{i_2},\ldots]]$ inductively by
\begin{flalign}
  \cAA_I &\= \cHH_I \in \frac{1}{z_i}R[[z_i]] \qquad \text{if $n = 1$,
and otherwise} \label{eq:calArec} \\ 
\cAA_{I} &\= \frac{1}{n-1} \sum_{1\leq r < s \leq n}\,
\sum_{k \geq 0} 
\sum_{\begin{smallmatrix} \bfp=(p_1,\ldots, p_k)\\ I=\{i_r,i_s\}\sqcup I_1\sqcup\cdots \sqcup I_k  
\end{smallmatrix}} \frac{1}{k!} \,\cHH_{i_r,i_s}^\bfp \cdot
\prod_{j=1}^k \cAA_{I_j}^{[p_j]}\,, \nonumber
\end{flalign}
where $\cAA_I^{[p]} := [z_i^p] \cAA_{I \cup \{i\}}$ for any $i \not\in I$.
We set $\mathcal{A}_{n}=\mathcal{A}_{[\![1,n]\!]}$. Note that $\cAA_I=\cHH_I$
by definition if $|I| =2$. We have 
chosen to sum in definition of $\cAA_{I}$ over all~$\bfp$ rather than
partitions $g=\sum_{i=1}^k g_i$ as e.g.\ in Theorem~\ref{intro:VolRec}. The two 
summations are equivalent, since~$g_i$ and~$p_i$ determine each other
once~$I$ has been partitioned. Our goal here is to show the following result.
\par
\begin{Thm}\label{BOintMV}
For all non-empty sets of positive integers $I$, we have $\cAA_I=\cHH_I.$
\end{Thm}
\par
For the proof of this theorem we will show that both $\cAA_I$ and $\cHH_I$ 
can be written as a sum which ranges over certain oriented trees
  (see Section~\ref{subsec:ori-tree}).
The two recursions can then be viewed as stemming from cutting the trees at a
local maximum (a ``top'') or a local minimum (a ``bottom'') respectively.

%%%%%%%%%%%%%%%%%%%%%
\subsection{Proof of Theorem~\ref{intro:VolRec} and
 Theorem~\ref{intro:IntFormula}} 
%%%%%%%%%%%%%%%%%%%%%%

We assume in this section that Theorem~\ref{BOintMV} holds and finish the proof of
Theorems~\ref{intro:VolRec} and~\ref{intro:IntFormula} under this assumption. We
abbreviate $\lambda = (\ell_1,\ldots,\ell_n)$ and recall from Section~\ref{sec:D2rec}
that we denoted the coefficients of $\mathcal{A}_n$ for $n \geq 2$ by
$$\mathcal{A}_n \= \mathcal{H}_n \= \sum_{\ell_1,\ldots, \ell_n \geq 1}
h_\lambda \, 
z_{i_1}^{\ell_1} \cdots z_{i_n}^{\ell_n} 
\,.$$
\par
\begin{Prop} The coefficients  $h_{\lambda}$
are uniquely determined 
by the recursion 
\ba \label{VRBO}
h_\lambda %\fraka(\lambda)
\= \sum_{1\leq r< s\leq n} \frac{\ell_r+ \ell_s}{n-1} \,
\sum_{k \geq 1} \frac{1}{k!} \, \sum_{\bfg, \bf\mu} 
h_{\proj^1}(\ell_r-1,\ell_s-1,\bfp) \cdot
\prod_{i=1}^k  h_{\lambda_i,p_i} 
\ea
for $n \geq 2$, where the summation is as in Theorem~\ref{intro:VolRec},
except that $\lambda_1\sqcup \cdots \sqcup \lambda_k = [\![1,n]\!]\smallsetminus\{r,s\}$. 
\end{Prop}
\par
\begin{proof} We begin by showing the formula for $\lambda$ with two entries,
which in view of~\eqref{eq:defhP1} is equivalent to show that
\begin{eqnarray*}
\sum_{\ell_1,\ell_2 \geq 1} \frac{h_{\ell_1,\ell_2}}
{\ell_1+\ell_2} z_1^{\ell_1}z_2^{\ell_2} &=& \sum_{k\geq 1}\frac{1}{k} \left(\sum_{p\geq 1}
h_p  z_1z_2\frac{z_1^p-z_2^p }{z_1-z_2} \right)^k\\
&=& - {\rm log} \left( \frac{z_1z_2}{z_1-z_2}\left(\cHH_1(z_2)-\cHH_1(z_1)\right)\right)\,.
\end{eqnarray*}
Since applying $(z_1 \frac{\partial}{\partial z_1}+z_2 \frac{\partial}{\partial z_2})$
to the left-hand side above gives $\cAA_2$, and since neither side has a constant term, 
this in turn follows from 
\begin{eqnarray*}
\cHH_2 \= \cAA_2
&=& -1+ \frac{z_1 \cHH_1'(z_1)-z_2 \cHH_1'(z_2)}{\cHH_1(z_2)- \cHH_1(z_1)}\\
&=& - \left(z_1 \frac{\partial}{\partial z_1}+z_2 \frac{\partial}{\partial z_2} \right)
{\rm log} \left( \frac{z_1z_2}{z_1-z_2}\left(\cHH_1(z_2)-\cHH_1(z_1)\right)\right)\,.
\end{eqnarray*}
\par 
For $\lambda$ with more entries, we deduce (for all $\ell_r,\ell_s \geq 1$ and 
all $\bfp$) from the preceding calculation that 
\begin{eqnarray*}
[z_r^{\ell_r}z_s^{\ell_s}] \cHH_{\{ r, s\}} &\=&  \sum_{k'>0} \frac{\ell_r+\ell_s}{k'!} \sum_{\bfp'} h_{\proj^1}(\ell_r-1,\ell_s-1,\bfp') \prod_{i=1}^{k'} h_{p_i'}\,\\
\text{and}\qquad [z_r^{\ell_r}z_s^{\ell_s}]\cHH_{\{ r, s\}}^\bfp &\=&  \sum_{k' > 0} \frac{\ell_r+\ell_s}{k'!} \sum_{\bfp'} h_{\proj^1}(\ell_r-1,\ell_s-1,\bfp\cup\bfp') \prod_{i=1}^{k'}
h_{p_i'}\,,
\end{eqnarray*}
Besides, the recursion formula~\eqref{eq:calArec} defining $\cAA_I$ can be translated
for~$\lambda$ with~$n$ parts into 
\begin{flalign} 
&\phantom{\=} h_\lambda 
\= \sum_{1\leq r< s\leq n}  \frac{1}{n-1}
\sum_{\begin{smallmatrix} k\geq 1,g_i, \lambda_i\\ |\lambda_i|>0 \end{smallmatrix}}
\frac{1}{k!} [z_r^{\ell_r}z_s^{\ell_s}] \cHH_{\{ r, s\}}^\bfp  \prod_{i=1}^k
h_{\lambda,p_i} 
 \label{eq:frakalambda} \\
&\= \sum_{1\leq r< s\leq n}  \frac{\ell_r+\ell_s}{n-1}
\sum_{k\geq 1, g_i, |\lambda_i|>0 \atop k' > 0, \bfp' }
\frac{1}{k!k'!} h_{\proj^1}(\ell_r-1,\ell_s-1,\bfp\cup\bfp')
\prod_{i=1}^k  h_{\lambda,p_i} 
\prod_{j=1}^{k'} h_{p_j'}
\,, \nonumber
\end{flalign} 
where $\lambda_1\sqcup \cdots \sqcup\lambda_k$ partitions 
$\lambda \setminus \{\ell_r, \ell_s \}$. Next we remark that the interior
sum of~\eqref{VRBO} is over all backbone graphs with the two markings labelled with $r$ and $s$ at the lower level.
In the preceding formula~\eqref{eq:frakalambda} the contribution of vertices
with at least one marking is separated from the vertices with no markings.
This choice results in a binomial coefficient  ${k+k' \choose k}$ and transforms
 $\frac{1}{k! k'!}$ into $\frac{1}{(k+k')!}$, thus showing that the two 
recursive formulas~\eqref{eq:frakalambda} and~\eqref{VRBO} are equivalent.
\end{proof}
\par
\begin{proof}[Proof of Theorem~\ref{intro:VolRec} and
Theorem~\ref{intro:IntFormula}]
For $\mu=(m_1,\ldots,m_n)$ consider the intersection numbers $a(\mu)$ that satisfy the recursion in
Theorem~\ref{thm:indintall}, 
and recall that $a(\mu) = a_i(\mu)$ is independent of the index $i$ by Proposition~\ref{independent}. 
In particular the $a(\mu)$ satisfy the recursion~\eqref{eq:ind2} for any distinguished pair of indices, and hence 
satisfy every weighted average of these recursions. We use the weighted
average where the recursion with $(i,j)$ distinguished is taken with weight
$\prod_{k \not\in\{i,j\}} (m_k+1)$. Conversely, the $a(\mu)$ are
uniquely determined by this weighted average and the initial values for $\mu$
of length one given in~\eqref{eq:ind1}.
\par
On the other hand, the collection of $(2g-2+n) \prod_{i=1}^n (m_i+1)a(\mu)$
and the collection of $h_{m_1+1,\ldots,m_n+1}$ 
both satisfy the recursion~\eqref{VRBO}, by observing that 
\be \label{eq:average}
(2g-2+n) a(\mu) \prod_{i=1}^n (m_i+1)
\= \sum_{1\leq r< s\leq n} \!\!\frac{(m_r+1+ m_s+1)}{n-1}  a(\mu) \prod_{i=1}^n (m_i+1)\,.
\ee
Note that $\cAA_1|_{h_{\ell}\mapsto \alpha_{\ell}} = \cAA$ by Theorem~\ref{thm:D2recursion} 
and since we already checked (see~\eqref{eq:fellinversion} and the subsequent proof) 
that the one-variable rescaled volumes $v(2g-2)$ and $a(2g-2)$ agree 
(see~\eqref{eq:volminviaa}) up to the factor $(2\pi i)^{2g}/(2g-1)!$.  This implies that 
\be \label{for:liftint}
a(\mu) \= \frac{h_{m_1+1,\ldots,m_n+1}|_{h_{\ell}\mapsto \alpha_{\ell}}}{(2g-2+n)
\prod_{i=1}^n (m_i+1)}\,.
\ee
The claim now follows from Theorem~\ref{thm:D2recursion}, Theorem~\ref{BOintMV}
and the conversion~\eqref{eq:volai} of volumes to the~$a(\mu)$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Oriented trees}
\label{subsec:ori-tree}
%%%%%%%%%%%%%%%%%%%%%%%%%%

We now start preparing for the proof of Theorem~\ref{BOintMV}. 
An {\em oriented tree} is the datum of a graph $G=(V, E\subset V\times V)$
whose underlying graph of $(V,E)$ is a tree. In particular it is
required to be connected. If $(v,v')\in E$, we will denote $v>v'$. Moreover,
a vertex $v\in V$ is called a {\em bottom} (respectively a {\em top}) if
there exists no $v'\in V$ such that $v > v'$ (respectively $v < v'$). We will
denote by $B(G)$ and $T(G)$ the sets of bottoms and tops of $G$. 
\par
For any oriented tree $G$ with $n$ vertices, we define the rational number
\be\label{eq:f(G)}
f^{\#}(G) :=\frac{ {\rm Card} \left\{ \sigma\colon V\overset{\sim}{\to} [\![1,n]\!],\ \text{s.t.} \forall (v,v')\in E, \sigma(v)>\sigma(v') \right\}}
{n!} \,,
\ee
whose numerator is the number of total orderings on the set of vertices
compatible with the orientation of~$G$.
\par
\begin{Lemma}\label{bottomtop}
The function $f^{\#}$ can be expressed as 
$$
f^{\#}(G)
\= \frac{1}{n} \cdot \sum_{v \in B(G)} \left(\prod_{G'} f^{\#}(G') \right)
\= \frac{1}{n} \cdot \sum_{v \in T(G)} \left(\prod_{G'} f^{\#}(G') \right)\,,
$$
where in both cases the product is over all connected components~$G'$
of the oriented graph obtained by deleting the vertex $v$.  
\end{Lemma}
\par
\begin{proof}
In order to define a total ordering on $V$ compatible with the orientation
of~$G$, we begin by choosing a minimal element $v\in V$. This element is
necessarily a bottom. Let us fix such a choice and denote by
$(G_1,\ldots,G_k)$ the connected components of $G \setminus \{v\}$.
Let $n_i$ be the number of vertices of $G_i$ for $1\leq i\leq k$.  A total
ordering on~$V$ with minimal element~$v$ is equivalent to choosing a total
ordering on the vertices of~$G_i$ for all $1\leq i\leq k$ and a partition
of $[\![1,n-1]\!]$ into $k$ sets of size $(n_1,\ldots,n_k)$. Such an ordering
on~$V$ is compatible with the orientation of $G$ if and only if each ordering
on the vertices of $G_i$ is compatible with the orientation of $G_i$ for all
$1\leq i\leq k$. This implies that the number of total orderings on $V$
with minimal element $v$ is equal to 
$$
\left(\begin{matrix} n-1 \\ n_1 \ \cdots \ \ n_k \end{matrix}\right) \cdot
\prod_{i=1}^k (n_i! f^{\#}(G_i))= (n-1)! \cdot \prod_{i=1}^k f^{\#}(G_i)\,.
$$
Summing over all possible choices of a minimal element, the number of total
orderings on vertices of $G$ compatible with the orientation of $G$ is equal to
$$
(n-1)! \cdot \sum_{v \in B(G)} \left(\prod_{G'} f^{\#}(G')
\right),
$$
which completes the proof.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Decorations of oriented trees}
%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $I$ be a non-empty finite set of positive integers. An {\em $I$-decoration} of an oriented tree $\Gamma=(V,E)$ is the datum of
a function ${\rm dec}\colon I\to V$ such that for each vertex~$v$ the number
of outgoing edges plus the number of decorations is equal to two, i.e.\
$$
\#({\rm dec}^{-1}(v)) + \#(E\cap (\{v\} \times V)) \=2\,
$$
for all $v\in V$. 
If $I$ has cardinality greater than one, we denote by ${\rm OT}(I)$ the set
of $I$-decorated oriented trees.  One can easily check that the following two
properties hold:
\begin{itemize}
\item if $I$ has cardinality $n\geq 2$, then $\Gamma$ has $n-1$ vertices;
\item a vertex of a decorated tree is a bottom if and only if it has exactly
  two markings. 
\end{itemize} 
\par
We denote by ${\rm OT}(I)^{v}, {\rm OT}(I)^{b}$ and ${\rm OT}(I)^{t}$ the sets
of decorated trees with a choice of an arbitrary vertex, a choice of a
bottom and a choice of a top, respectively. 
If~$I = \{i\}$ has only one element, we define ${\rm OT}(\{i\})^{v}=\{i\}$ as a trivial graph decorated by~$i$. 
\par
\begin{Lemma}\label{le:phi-psi}
If $I$ has cardinality greater than one, then there is a bijection
\begin{eqnarray}\label{bijection1}
  \varphi^{t}\colon {\rm OT}(I)^{t} \to \left(\bigcup_{I'\subset I} {\rm OT}(I')^{v}
  \times {\rm OT}(I\setminus I')^{v}\right)\bigg/ (I' \sim I\setminus I')\,
\end{eqnarray}
given by cutting at a top vertex, were the union is over all non-empty
proper subsets. Similarly, there is a bijection
\begin{eqnarray}\label{bijection2}
\varphi^{b}\colon  {\rm OT}(I)^{b} \to \bigcup_{\{i_1,i_2\}\subset I, k > 0} \left(  \bigcup_{I=\{i_1,i_2\} \sqcup  I_1\sqcup \cdots \sqcup I_k} \prod_{j=1}^k {\rm OT}(I_j\cup \{e_j\} ) \right)\bigg/S_k\,
\end{eqnarray}
given by cutting at a bottom vertex, were the union is over all partitions of
$I$ into $k+1$ non-empty sets such that the first distinguished set has 
precisely two elements and where the element $e_j={\rm max}(I)+j$
for all $1\leq j\leq k$.
\par
We denote by $\psi^t$ and $\psi^b$
the inverses of $\varphi^t$ and $\varphi^b$, respectively.
\end{Lemma}
\par
\begin{proof}
Given a decorated tree with a chosen top vertex $v$, we define its image
under $\varphi^{t}$ as follows:
\begin{itemize}
\item If there are two markings on $v$, then $v$ has no outgoing edges, hence the graph has $v$ as a unique vertex and 
$I$ has only two elements. It follows that ${\rm OT}(I)^{v}$ has only one element (and so does the right-hand side
of~\eqref{bijection1}). 
\item If there is only one marking $i\in I$ on $v$ and one outgoing edge to a vertex $v'<v$, then $I'=\{i\}$ and the corresponding element in ${\rm OT}(I\setminus I')^{v}$ is the graph obtained by deleting $v$ and choosing $v'$ as the distinguished vertex.
\item If there are two outgoing edges to vertices $v'<v$ and $v''<v$, then $v$ has no $I$-markings and the graph obtained by deleting $v$ has two connected components. We define $I'$ to be the set of markings
on the component containing $v'$ and define the corresponding elements of ${\rm OT}(I')^{v}$
and ${\rm OT}(I\setminus I')^{v}$ to be the connected components containing~$v'$
and~$v''$ as chosen vertices, respectively. 
\end{itemize}
The inverse of $\varphi^t$ in the first two cases is clear, and in the last case is given by adding a top vertex adjacent to the two chosen vertices.
\par
Given a decorated tree with a chosen bottom vertex~$v$, in the same spirit we define the function~$\varphi^b$ as follows. Since $v$ is a bottom, it has no outgoing edges, hence it has exactly two $I$-markings~$i_1$ and $i_2$, and the corresponding $k$ graphs on the right-hand side of~\eqref{bijection2} are the $k$ connected components of the graph obtained by removing $v$. The inverse of $\varphi^{b}$ is given by gluing these $k$ graphs back to $v$ along the vertices marked by $e_1, \ldots, e_k$.  
\end{proof}
\par
Now we fix a ring $R'$ and a function $g\colon {\rm OT}(I)\to R'$. By slight abuse
of notation, we write $f^{\#}\colon {\rm OT}(I)\to \QQ$ for the composition of $f^{\#}$ defined in~\eqref{eq:f(G)} with the forgetful map of the decorations.  As a consequence of the two
preceding lemmas we see that the sum 
$$
S(g)=\sum_{\Gamma \in {\rm OT}(I)} f^{\#}(\Gamma) g(\Gamma)
$$
can be rewritten in two different ways, namely
\be \label{eq:rew1}
S(g)=\frac{1}{2(n-1)} \sum_{I'\subset I} \sum_{\begin{smallmatrix} (\Gamma', v') \in {\rm OT}(I')^v \\ (\Gamma'', v'') \in {\rm OT}(I\setminus I')^v  \end{smallmatrix}}
f^{\#}(\Gamma') f^{\#}(\Gamma'') \,g\bigl(\psi^{t}(\Gamma', v', \Gamma'', v'' )
\bigr) 
\ee
and  
\be \label{eq:rew2}
S(g)=\frac{1}{n-1} \sum_{\begin{smallmatrix} \{i_1,i_2\}\subset I, k > 0\\ I=\{i_1,i_2\} \sqcup  I_1\sqcup \cdots \sqcup I_k\end{smallmatrix}} \!\!\frac{1}{k!}
\sum_{(\Gamma_j)_{j=1}^k}   \Bigl(\prod_{j=1}^k f^{\#}(\Gamma_j)\Bigr)
\cdot g\Bigl(\psi^{b}\Bigl(\prod_{j=1}^k \Gamma_j\Bigr)\Bigr) \,,
\ee
where $n$ is the cardinality of $I$ (i.e.~$\Gamma \in {\rm OT}(I)$ has $n-1$ vertices as remarked before).  

%%%%%%%%%%%%%%%%
\subsection{Explicit expansions over decorated trees}
%%%%%%%%%%%%%%%%%

In order to finish the proof of Theorem~\ref{BOintMV}, we will show that both $\cAA_I$ and $\cHH_I$ are equal to a generating series $\cSS_I$ that is directly defined as a sum over ${\rm OT}(I)$. 
\par
Let $\Gamma=(V,E, I\to V)$ be an oriented tree with decoration by $I$. A {\em twist assignment}
on $\Gamma$ is a function $\bfp\colon E \to \ZZ_{>0}$. We work over the ring
$R[[(z_{i})_{i\in I}, (z_{e})_{e\in E}]]$ of formal series in variables indexed by $I\cup E$.
Given a twisted decorated oriented tree, we define the contribution of a vertex as
 $$
\cHH_v= \cHH_{2}^{\bfp_v}(z_{v,1},z_{v,2}) \in R[[(z_{i})_{i\in I}, (z_{e})_{e\in E}]]\,,
$$
where $\bfp_v$ is the list of twists associated to all vertices~$v'$ with $v'>v$ and
where $(z_{v,1},z_{v,2})$ are the variables attached to either the markings of~$v$ or
the outgoing edges from $v$ to vertices~$v'$ with $v'<v$. Then we define the contribution of the
oriented tree $\Gamma$ as
$$
{\rm cont}(\Gamma) \= \sum_{\bfp: E\to \ZZ_{>0}} \bigl[\prod_{e\in E}
z_{e}^{\bfp(e)}\bigr] \prod_{v\in V} \cHH_{v}\,
$$
if $\Gamma$ is non-trivial, and define ${\rm cont}(\Gamma) = \cHH_v$ for the trivial graph $\Gamma$ with a unique vertex $v$ and no edges.  
Finally, we set $\cSS_{\{i\}} = \cHH_{\{i\}} = \cAA_{\{i\}}$ and for $|I| \geq 2$
$$
\cSS_{I} \= \sum_{\Gamma\in {\rm OT}(I)} f^{\#}(\Gamma) \,{\rm cont}(\Gamma)\,.
$$
\par
\begin{proof}[End of the proof of Theorem~\ref{BOintMV}]
We will show that $\cSS_{I}=\cHH_{I}$ and $\cSS_{I}=\cAA_{I}$ for all sets of positive
integers~$I$ with $n={\rm Card}(I)>2$. The equalities in the case $n=2$ are obvious 
from the definition.
We assume now that $n\geq 3$ and that $\cSS_{I'}=\cHH_{I'}=\cAA_{I'}$ for
all~$I'$ such that ${\rm Card}(I')<n$.
\par
We first prove that $\cSS_{I}=\cHH_{I}$. We begin by rewriting the defining
equation~\eqref{eq:Hindform} with two auxiliary ``edge'' variables
$z_{e'}$ and $z_{e''}$ for distinct indices $e', e'' \in \NN \setminus I$ as 
\bes
\cHH_I \= \frac{1}{2(n-1)} \sum_{I'\subset I}\sum_{p_{e'},p_{e''}>0} \left( [z_{e'}^{p_{e'}}z_{e''}^{p_{e''}}] \cHH_{\{e',e''\}}  \right)  \frac{\partial\cHH_{I'}}{\partial h_{p_{e'}}}
\frac{\partial \cHH_{I\setminus I'}}{\partial h_{p_{e''}}} \,.
\ees
To evaluate the derivative of $\cHH_{I'}$, there are two cases to consider, depending
on the cardinality of~$I'$. If $I'=\{i\}$, then
$\frac{\partial\cHH_{I'}}{\partial h_{p}}=z_{i}^{p}$ for all $p>0$. Otherwise, we use
the induction hypothesis to compute that 
\begin{eqnarray*}
\frac{\partial\cHH_{I'}}{\partial h_{p}}\=\frac{\partial\cSS_{I'}}{\partial h_{p}}&=& \sum_{\Gamma\in {\rm OT}(I')} f^{\#}(\Gamma) \frac{\partial\ {\rm cont}(\Gamma)}{\partial h_{p}} \\
&=& \sum_{\Gamma\in {\rm OT}(I')} f^{\#}(\Gamma) \sum_{\bfp: E\to \ZZ_{>0}} \left[\prod_{e\in E} z_{e}^{\bfp(e)}\right ] \sum_{v\in V} \frac{\partial\cHH_{v}}{\partial h_{p}} \left( \prod_{\hat{v}\neq v} \cHH_{\hat{v}}\right)\\
&=&\sum_{(\Gamma,v) \in {\rm OT}(I')^v} f^{\#}(\Gamma) \sum_{\bfp: E\to \ZZ_{>0}} \left[\prod_{e\in E} z_{e}^{\bfp(e)}\right ] \frac{\partial\cHH_{v}}{\partial h_{p}} \left( \prod_{\hat{v}\neq v} \cHH_{\hat{v}}\right)\,.
\end{eqnarray*}
Now we assume that both $I'$ and $I\setminus I'$ have at least two elements. Take two
oriented trees $(\Gamma'=(E',V',I'\to V'),v') \in {\rm OT}(I')^v$ and
$\Gamma''=(E'',V'',I\setminus I'\to V''),v'') \in {\rm OT}(I'')^v$. Let $(\Gamma=(V,E,I\to V) ,v)=\psi^t(\Gamma',\Gamma'')$ be the combined graph in ${\rm OT}(I)^{t}$ as described in Lemma~\ref{le:phi-psi}. 
\par
The datum of two twist assignments~$\bfp'$ and $\bfp''$ on $\Gamma'$ and $\Gamma''$ respectively 
together with a pair of positive integers $(p_{e'}, p_{e''})$ is equivalent to the datum of a twist assignment
$\bfp\colon V\to \ZZ_{>0}$ on the graph~$\Gamma$. Moreover, the contributions of the
vertices of~$\Gamma$ with the twist assignment~$\bfp$ are given by
\begin{itemize}
\item $ \cHH_{\{e',e''\}}$ for the top vertex $v$,
\item $\frac{\partial\cHH_{v'}}{\partial h_{p_{e'}}}$ and $\frac{\partial\cHH_{v''}}{\partial h_{p_{e''}}}$ for the two distinguished vertices of $\Gamma'$ and $\Gamma''$, and 
\item $\cHH_{\hat{v}}$ for all other vertices $\hat{v}$.
\end{itemize}
One checks that this is still true if one of the graphs $\Gamma'$ or $\Gamma''$
has only one marking. In summary we obtain that 
\begin{eqnarray*}
\cHH_I&=& \frac{1}{2(n-1)} \sum_{I'\subset I}  \sum_{\begin{smallmatrix} (\Gamma',v')\in {\rm OT}(I')^v \\ (\Gamma'',v'')\in {\rm OT}(I\setminus I')^v\end{smallmatrix}}f^{\#}(\Gamma')f^{\#}(\Gamma'') \sum_{\bfp\colon E\to \ZZ_{>0}} \left[z_{e}^{\bfp(e)}\right] \left(\prod_{\hat{v}\in V} \cHH_{\hat{v}} \right) \\
&=&\frac{1}{2(n-1)} \sum_{I'\subset I}  \sum_{\begin{smallmatrix} (\Gamma',v')\in {\rm OT}(I')^v \\ (\Gamma'',v'')\in {\rm OT}(I\setminus I')^v\end{smallmatrix}}f^{\#}(\Gamma')f^{\#}(\Gamma'')
{\rm cont}(\psi^t (\Gamma',v',\Gamma'',v''))\\
&=& \cSS_I\,,
\end{eqnarray*}
where we use~\eqref{eq:rew1} to pass from the above second line to the third.
\par
\medskip
Finally, we prove that $\cSS_{I}=\cAA_{I}$ by a similar argument. It suffices to prove for the case $I = [[1, n]]$.  
Using the induction hypothesis that $\cAA_{I_j\cup \{n+j\}} = \cSS_{I_j\cup \{n+j\}}$, we can rewrite the inductive definition~\eqref{eq:calArec} of $\cAA_I$ as   
\bas
 \cAA_{n}  &\= \frac{1}{(n-1)} \sum_{\begin{smallmatrix} k > 0,  [\![1,n]\!] =\{i_1,i_2\}\sqcup I_1\sqcup\cdots \sqcup I_k  \\ \Gamma_j \in {\rm OT}(I_j\cup\{n+j\})
\end{smallmatrix}}  \!\!\!\!  \!\!\!\! \frac{ \prod_{j=1}^k f^{\#}(\Gamma_j)}{k!}\,  
\\
&\phantom{\=} \quad \cdot
\sum_{\bfp = (p_1,\ldots,p_k) \atop
\bfp^{(j)}\colon E_j\to \ZZ_{>0}}
\Big([z_{n+1}^{p_1}\cdots z_{n+k}^{p_k}]\, \cHH_{i_1,i_2}^{\bfp}\Big) \left[\prod_{j=1}^k\prod_{e\in E_j} z_e^{\bfp^{(j)(e)}}\right]  \prod_{v\in V_1\cup \cdots \cup V_k} \cHH_v 
\,.
\eas
The datum of $\bfp$ together with the twist assignments $\bfp^{(j)}$ for the
split graphs~$\Gamma_j$ for $1\leq j\leq k$ is equivalent to a twist assignment for the
combined graph $\Gamma = \psi^b(\Gamma_1,\ldots, \Gamma_k)$ defined in Lemma~\ref{le:phi-psi}. Moreover, given such a twist assignment the contribution
of the vertex carrying~$i_1$ and~$i_2$ is $\cHH_{i_1,i_2}^{\bfp}$. Thus we obtain that 
\begin{eqnarray*}
\cAA_{n} \!\!\!\! &=&  \!\!\!\!   \frac{1}{(n-1)}  \sum_{\begin{smallmatrix} k > 0,\\
[\![1,n]\!] =\{i_1,i_2\}\sqcup I_1\sqcup \cdots \sqcup I_k  \\ \Gamma_j \in {\rm OT}(I_j\cup\{n+j\})
\end{smallmatrix}}  \!\!\!\!  \!\!\!\! \frac{ \prod_{j=1}^k f^{\#}(\Gamma_j)}{k!}  
\cdot  \sum_{\bfp\colon E(\Gamma) \to \ZZ_{>0}} \Big[\prod_{e\in E(\Gamma)} z_e^{\bfp(e)}\Big]  \prod_{v\in V(\Gamma)} \cHH_v \\
&=&  \!\!\!\!  \frac{1}{(n-1)}  \sum_{\begin{smallmatrix} k > 0, \\  
[\![1,n]\!] =\{i_1,i_2\}\sqcup I_1\sqcup \cdots \sqcup I_k  \\ \Gamma_j \in {\rm OT}(I_j\cup\{n+j\})
\end{smallmatrix}}  \!\!\!\!  \!\!\!\! \frac{ \prod_{j=1}^k f^{\#}(\Gamma_j)}{k!}  \cdot {\rm cont}(\psi^b(\Gamma_1,\ldots, \Gamma_k))\\
&=& \cSS(I)\,,
\end{eqnarray*}
where we use~\eqref{eq:rew2} to pass from the above second line to the third.
\end{proof}



