via the restriction of the map $\Psi\restr_{[\tau,\hat{1}_n]}$ from Proposition <ref>. The number $t=\abs{\alpha}$ is equal to $\abs{\tau}-\asc_\eta(\tau)$, the integers $p_j$ are the cardinalities of $A_j$ and $\sigma_{\mathrm{max}}^{\mathrm{asc}}(\tau,\eta)$ is the ordered set partition corresponding to $\lambda =\alpha$.

Before discussing the question of mixed cumulants we prove an analogue of
Weisner's lemma for ordered set partitions.

*  For $\tau,\eta \in \OSP_n$, we have

\begin{equation}
\label{eq:weisnercoeff}
\begin{split}
w(\tau,\eta)&:=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \curlywedge \eta=\tau}}\widetilde{\mu}(\sigma,\hat{1}_n)\\
\begin{cases}\displaystyle
\int_{-1}^0 x^{\abs{\tau} -\asc_\eta(\tau)-1}(1+x)^{\asc_\eta(\tau)}\,dx=\frac{(-1)^{\abs{\tau}-\asc_\eta(\tau)-1}}{\abs{\tau}\binom{\abs{\tau}-1}{\asc_\eta(\tau)}}, &\bar{\tau} \leq \bar{\eta},\\
0, & \bar{\tau} \not\leq \bar{\eta}.
\end{cases}
\end{split}
\end{equation}

*  More generally, for $\tau,\eta,\pi \in \OSP_n$, we have

\[
w(\tau,\eta,\pi):=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \curlywedge \eta=\tau\\ \sigma \leq \pi}}\widetilde{\mu}(\sigma,\pi)=
\begin{cases}\displaystyle
\prod_{P\in\pi} w(\tau\restr_P,\eta\restr_P), &\bar{\tau} \leq \bar{\eta},~\tau \leq \pi,\\
0, & \text{otherwise}.
\end{cases}
\]

We call the numbers $w(\tau,\eta,\pi)$ and
$w(\tau,\eta)=w(\tau,\eta,\hat{1}_n)$ Weisner coefficients.

(<ref>)   If $\bar{\tau} \not\leq \bar{\eta}$, then there is no
$\sigma$ such that $\sigma \curlywedge \eta=\tau$ and the sum is empty.
Let us therefore assume henceforth that $\bar{\tau} \leq \bar{\eta}$.
We take up the end of the proof of Lemma <ref> where we
established the poset isomorphism

\begin{equation}
\begin{split}
\{\sigma \in\OSP_n \mid \sigma \curlywedge \eta=\tau \} &
\cong\IP_{p_1} \times \IP_{p_2} \times \cdots \times \IP_{p_t} \\
& \cong \Bool_{p_1-1} \times \Bool_{p_2-1} \times \cdots \times \Bool_{p_t-1} \\
& \cong \Bool_{p_1+\cdots+p_t   -t},
\end{split}
\end{equation}

where the second isomorphism follows from Proposition <ref> and $p_i$ denotes the length of the $i$-th ascending run.
The latter contains $p_i-1$ rises and therefore the total number of
ascents is
$\asc_\eta(\tau) = p_1+p_2+\cdots +p_t-t$.
In the identification above, a partition $\sigma$ is mapped to a subset
$A\subseteq [p_1+p_2+\cdots+p_t-t]$
with  $\abs{A} = \abs{\sigma}-t$ elements and we have

\begin{equation}
\widetilde{\mu}(\sigma,\hat{1}_n)= \frac{(-1)^{\abs{\sigma}-1}}{\abs{\sigma}} = \frac{(-1)^{\abs{A}+t-1}}{\abs{A}+t}.
\end{equation}

Performing the sum we obtain

\begin{equation}
\begin{split}
\sum_{\substack{\sigma \in \OSP_n \\ \sigma \curlywedge \eta=\tau}}\widetilde{\mu}(\sigma,\hat{1}_n)
&=  \sum_{A \subseteq [\asc_\eta(\tau)]}\frac{(-1)^{\abs{A}+t-1}}{\abs{A}+t}\\
&= \sum_{k=0}^{\asc_\eta(\tau)}\binom{\asc_\eta(\tau)}{k} \frac{(-1)^{k+t-1}}{k+t} \\
&= \int_{-1}^0 \sum_{k=0}^{\asc_\eta(\tau)}\binom{\asc_\eta(\tau)}{k} x^{k+t-1}\,dx \\
&= \int_{-1}^0 x^{t-1} (1+x)^{\asc_\eta(\tau)}\,dx \\
&= (-1)^{t-1}B(t, \asc_\eta(\tau)+1),
\end{split}
\end{equation}

where $B$ is the beta function $B(a,b)= \frac{\Gamma(a )\Gamma(b )}{\Gamma(a+b)}$
which can be written in terms of binomial coefficients as desired.

(<ref>)   In order for the set $\{\sigma\in\OSP_n \mid \sigma \leq \pi,~ \sigma
\curlywedge \eta=\tau\}$ to be nonempty,
it is necessary that $\tau \leq \pi$ and $\bar{\tau}\leq \bar{\eta}$. We adapt
the notations from Definition <ref>
and infer from Proposition <ref> that there exists an
interval partition $\rho=(R_1,R_2,\dots,R_p) \in \IP_{s}$ such that

\begin{equation}
\pi = (P_1,P_2,\dots,P_p)= (\bigcup_{i\in R_1} E_{m_i,n_i},\bigcup_{i\in R_2} E_{m_i,n_i}, \dots, \bigcup_{i\in R_p} E_{m_i,n_i}).
\end{equation}

If follows from Lemma <ref> that $\sigma$ belongs to $[\tau,
\sigma_{\max{}}^{\mathrm{asc}}(\tau,\eta)]$.
In addition, $\sigma$ must satisfy $\sigma \leq \pi$. Hence, the ascending runs considered in Lemma <ref> are split by the blocks of $\pi$.
More precisely, for each $k\in[p]$ we decompose $(m_i)_{i\in R_k}$ into
ascending runs, which give rise to the interval partition
$\gamma_k=(G_{k,1},\dots, G_{k,u_k}) \in \IP_{R_k}$ where $G_{k,j}$
consists of the indices $i$ of the $j$-th ascending run of $(m_i)_{i\in R_k}$.

\begin{equation}
\sigma_{\max{}}^{\mathrm{asc}}(\tau\restr_{P_k},\eta\restr_{P_k}):= (\bigcup_{i\in G_{k,1}} E_{m_i,n_i},\dots, \bigcup_{i\in G_{k,u_k}} E_{m_i,n_i}).
\end{equation}

and for each $P\in\pi$, we pick an arbitrary $\sigma_P \in\OSP_P$ from the interval
$[\tau\restr_P, \sigma_{\max{}}^{\mathrm{asc}}(\tau\restr_P,\eta\restr_P)]$,
concatenate them and obtain
$\sigma=\sigma_{P_1} \sigma_{P_2}\cdots\sigma_{P_p} \in \OSP_n$.
Since $\widetilde{\mu}(\sigma,\pi)$ is the product of
$\frac{(-1)^{\sharp(\sigma\restr_P)-1}}{\sharp(\sigma\restr_P)}$ over
$P\in\pi$, the conclusion follows.

Examples of the Weisner coefficients will be given in Example <ref>.
we are now ready to proceed with the investigation of cumulants with independent entries.

Let $\pi,\eta\in\OSP_n$ and
$(i_1,i_2,\ldots,i_n) \in \N^n$  be a tuple with kernel $\kappa(i_1,i_2,\ldots,i_n)=\eta$. Then

K_\pi(X_1^{(i_1)}, X_2^{(i_2)}, \ldots, X_n^{(i_n)})
= \sum_{\tau \in \OSP_n}
\varphi_\tau(X_1,X_2,\ldots,X_n) \, w(\tau,\eta,\pi)

We proceed as in the proof of Proposition <ref>.
Expressing cumulants in terms of moments (see Theorem <ref>), we have

\begin{equation}\label{eq119}
\begin{split}
K_\pi(X_1^{(i_1)},X_2^{(i_2)}, \ldots, X_n^{(i_n)})
&= \sum_{\substack{\sigma \in \OSP_n \\ \sigma \leq \pi}}
\varphi_\sigma(X_1^{(i_1)}, X_2^{(i_2)},\ldots,X_n^{(i_n)})\,\widetilde{\mu}(\sigma,\pi) \\
&=  \sum_{\substack{\sigma \in \OSP_n \\ \sigma \leq \pi}} \varphi_{\sigma\curlywedge\eta}(X_1,X_2,\ldots,X_n)\, \widetilde{\mu}(\sigma,\pi)  \\
&=\sum_{\tau \in \OSP_n}  \varphi_\tau(X_1,X_2,\ldots,X_n)\left(\sum_{\substack{\sigma \in \OSP_n \\ \sigma\curlywedge\eta =\tau, \sigma \leq \pi }} \widetilde{\mu}(\sigma,\pi) \right) \\
&= \sum_{\tau \in \OSP_n} \varphi_\tau(X_1,X_2,\ldots,X_n) \, w(\tau,\eta,\pi).
\end{split}
\end{equation}

Since $w(\tau,\eta,\pi) \neq0$ for $\bar{\tau} \leq \bar{\eta}, \tau \leq \pi$
and in general the expectation values $\varphi_\tau(X_1,X_2,\dots,X_n)$
among different $\tau$ do not cancel each other, we cannot expect the vanishing of cumulants without further assumptions.
However we can express cumulants with independent entries also in terms of cumulants of
lower orders. In this case it turns out that the coefficients are determined by the number of plateaux.

Let $\eta, \tau \in\OSP_n$ such that $\bar{\tau} \leq \bar{\eta}$, then the
restriction of the map $\Psi\restr_{[\tau,\hat{1}_n]}$ from
Proposition <ref> establishes a poset isomorphism

\begin{equation}\label{iso flat}
\{\sigma \in\OSP_n \mid \sigma \geq \tau, \bar{\sigma} \leq \bar{\eta} \}  \to
\IP_{q_1} \times  \IP_{q_2} \times \cdots \times \IP_{q_r},
\end{equation}

where $1 \leq q_i \leq n$ are the lengths of the level runs and
the number $r$ is equal to $\abs{\tau}-\plat_\eta(\tau)$.
In particular, there is an ordered set partition $\sigma_{\max{}}^{\mathrm{pla}}(\tau,\eta)$ such that

\{\sigma \in\OSP_n \mid \sigma \geq \tau, \bar{\sigma} \leq \bar{\eta} \}  =[\tau, \sigma_{\max{}}^{\mathrm{pla}}(\tau,\eta)].

The proof is similar to that of Lemma <ref>.
Write $\tau=(E_{m_1,n_1},E_{m_2,n_2}, \ldots, E_{m_s, n_s})$ and
$\eta=(E_1,E_2,\ldots,E_e)$ as in Definition <ref>.
Let $\sigma \geq \tau$, then from Proposition <ref> we infer
that there is an interval partition
$\lambda=(L_1,L_2,\ldots, L_l) \in \IP_{s}$ such that

\begin{equation}
\sigma=(\bigcup_{i\in L_1} E_{m_i,n_i}, \ldots, \bigcup_{i\in L_l} E_{m_i,n_i}).
\end{equation}

Let $(m_i)_{i=1}^{i_1}, (m_i)_{i=i_1+1}^{i_2},\ldots, (m_i)_{i=i_{r-1}+1}^s$ be
the decomposition of $(m_i)_{i=1}^s$ into level runs. This decomposition
determines interval blocks $B_j:= (i_{j-1}+1,i_{j-1}+2,\ldots, i_j)$
($i_0=0,i_r=s$) and hence gives rise to an interval partition $\beta =
(B_1,B_2,\ldots,B_r)$. In order that $\bar{\sigma} \leq \bar{\eta}$, each $L_i$
connects only plateaux, which is equivalent to the condition that $\lambda\leq
\beta$.
Denoting by $q_j=\abs{B_j}=i_j-i_{j-1}$, we get the isomorphism (<ref>).  The ordered
set partition $\sigma_{\max}^{\plat}(\tau,\eta)$ corresponds to the
choice $\lambda=\beta$.

*  For $\tau,\eta \in \OSP_n$, we have

\[
\begin{split}
\gold(\tau,\eta)&:=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \geq \tau}}\widetilde{\zeta}(\tau,\sigma)\,w(\sigma,\eta)\\
\begin{cases}\displaystyle
\frac{1}{q_1! q_2! \cdots q_r!}\int_{-1}^0 x^{\des_\eta(\tau)}(1+x)^{\asc_\eta(\tau)}\prod_{j=1}^r P_{q_j}(x)\,dx, &\bar{\tau} \leq \bar{\eta},\\
0, & \bar{\tau} \not\leq \bar{\eta},
\end{cases}
\end{split}
\]

where $q_j$ are the integers in Lemma <ref> and

P_q(x)=\sum_{k=1}^q k! \, S(q,k)\, x^{k-1} = E_q(x,x+1)

where $S(q,k)$ are the Stirling numbers of the second kind
and $E_n(x,y)=\sum_{\sigma\in\SG_n}x^{\des\sigma}y^{\asc\sigma}$ are the
homogeneous Eulerian polynomials
The first few polynomials are

\[
P_1(x)=1, \quad P_2(x)=2x +1,\quad P_3(x) = 6x^2 + 6x +1,\quad\dots
\]

More generally, for $\tau,\eta,\pi \in \OSP_n$, we have

\[
\gold(\tau,\eta,\pi):=\sum_{\substack{\sigma \in \OSP_n \\  \sigma \in[\tau, \pi]}}\widetilde{\zeta}(\tau,\sigma)\,w(\sigma,\eta,\pi)=
\begin{cases}\displaystyle
\prod_{P\in\pi} \gold(\tau\restr_P,\eta\restr_P), &\bar{\tau} \leq \bar{\eta},~\tau \leq \pi,\\
0, & \text{otherwise}.
\end{cases}
\]

The name “Goldberg coefficients” originates from the
Campbell-Baker-Hausdorff formula, see Section <ref> below,
in particular Theorem <ref>.
Some examples of $g(\tau,\eta)$ will be computed in Example <ref>.

The expansion of $P_q(x)$ in terms of Stirling coefficients was first proved
by Frobenius <cit.>.

If $\bar{\tau} \not\leq \bar{\eta}$, then $w(\sigma,\eta)=0$ for all $\sigma \geq \tau$ and so $\gold(\tau,\eta)=0$. Assume hereafter that $\bar{\tau} \leq \bar{\eta}$.
From Lemma <ref> we have the isomorphism

\begin{equation}
\begin{split}
\{\sigma \in\OSP_n \mid \sigma \geq \tau, \bar{\sigma} \leq \bar{\eta} \}
&\cong \IP_{q_1} \times \IP_{q_2} \times \cdots \times \IP_{q_r}, \\
\sigma &\mapsto (\sigma_1,\sigma_2,\ldots,\sigma_r).
\end{split}
\end{equation}

Since $r=\abs{\tau}-\plat_\eta(\tau)=\des_\eta(\tau)+\asc_\eta(\tau)+1$, we
have $\abs{\sigma}-\asc_\eta(\tau)-1 = \sum_{i=1}^r
(\abs{\sigma_i}-1)+\des_\eta(\tau)$. Note also that $[\tau:\sigma]!
=\prod_{i=1}^r \prod_{S\in\sigma_i}\abs{S}!$.  Since $\sigma$ just connects the
blocks of a level run of $(m_i)_{i=1}^{\abs{\tau}}$, it does not change the
number of ascents: $\asc_\eta(\sigma)=\asc_\eta(\tau)$ and we have

\begin{equation}
\begin{split}
&\sum_{\substack{\sigma \in \OSP_n \\ \sigma \geq \tau}}
\widetilde{\zeta}(\tau,\sigma)
\,
w(\sigma,\eta) \\
&\qquad\qquad=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \geq \tau, \bar{\sigma} \leq \bar{\eta}}}\frac{1}{[\tau:\sigma]!} \int_{-1}^0 x^{\abs{\sigma} -\asc_\eta(\tau)-1}(1+x)^{\asc_\eta(\tau)}\,dx \\
&\qquad\qquad=  \sum_{(\sigma_1,\sigma_2,\ldots,\sigma_r) \in\IP_{q_1} \times\IP_{q_2} \times \cdots \times \IP_{q_r}} \int_{-1}^0 x^{\des_\eta(\tau)}(1+x)^{\asc_\eta(\tau)}\prod_{i=1}^r\frac{x^{\abs{\sigma_i}-1}}{\prod_{S\in\sigma_i }\abs{S}!} dx \\
&\qquad\qquad=   \int_{-1}^0 x^{\des_\eta(\tau)}(1+x)^{\asc_\eta(\tau)}\prod_{i=1}^r\left(\sum_{\rho \in \IP_{q_i}}\frac{x^{\abs{\rho}-1}}{\prod_{R\in\rho}\abs{R}!}\right) dx.
\end{split}
\end{equation}

For $q\in\N$ we have

\begin{equation}
\sum_{\rho \in \IP_q}\frac{x^{\abs{\rho}-1}}{\prod_{R\in\rho}\abs{R}!} = \sum_{\substack{n_1+\cdots+n_k=q\\ n_i\geq1, k\geq1}}\frac{x^{k-1}}{n_1! \cdots n_k!}.
\end{equation}

For each fixed $k$, the sum $\sum_{\substack{n_1+\cdots+n_k=q\\
n_i\geq1}}\frac{q!}{n_1! \cdots n_k!}$ is the number of ways of
distributing $q$ distinct objects among $k$ nonempty urns, so it equals  $k!
S(q,k)$ and the proof is complete.

(<ref>)   The idea of the proof is similar to Proposition <ref>(<ref>) and we omit the proof.

Let $\pi,\eta\in\OSP_n$.
Suppose that a tuple $(i_1,i_2,\ldots,i_n)$ has kernel $\kappa(i_1,i_2,\ldots,i_n)=\eta$. Then

K_\pi(X_1^{(i_1)},X_2^{(i_2)}, \ldots, X_n^{(i_n)}) = \sum_{\tau \in \OSP_n} K_\tau(X_1,X_2,\ldots,X_n) \, \gold(\tau,\eta,\pi).

Using Proposition <ref>, we have

\begin{equation}
\begin{split}
K_\pi(X_1^{(i_1)},X_2^{(i_2)}, \ldots, X_n^{(i_n)})
&= \sum_{\sigma \in \OSP_n} \varphi_\sigma(X_1,X_2,\ldots,X_n) \, w(\sigma,\eta,\pi) \\
&=  \sum_{\sigma \in \OSP_n} \sum_{\substack{\tau \in\OSP_n\\ \tau \leq \sigma}}K_\tau(X_1,X_2,\ldots,X_n) \, \widetilde{\zeta}(\tau,\sigma)\,w(\sigma,\eta,\pi) \\
&=  \sum_{\tau \in \OSP_n} K_\tau(X_1,X_2,\ldots,X_n) \left(\sum_{\substack{\sigma \in\OSP_n\\ \sigma \in[ \tau,\pi]}}\widetilde{\zeta}(\tau,\sigma)\,w(\sigma,\eta,\pi)\right) \\
&=  \sum_{\tau \in \OSP_n} K_\tau(X_1,X_2,\ldots,X_n)\,\gold(\tau,\eta,\pi).
\end{split}
\end{equation}

For the third equality, we use Proposition <ref>(<ref>)
which asserts that $w(\sigma,\eta,\pi)=0$ if  $\sigma \not\leq \pi$.

We can characterize $\cS$-independence in terms of the above proposition, that is “semi-vanishing” of mixed cumulants.

Let $\cS=(\alg{U},\tilde{\phi})$ be a spreadability system for a given
ncs $(\alg{A},\phi)$. A sequence of subalgebras $(\alg{A}_i)_{i\in I}$ of $\alg{A}$, where
$I\subseteq \N$, is $\cS$-independent if and only if for any tuple $(i_1,i_2,\dots,i_n)\in I^n$, any random variables $(X_1,X_2,\dots,X_n)\in \alg{A}_{i_1} \times \alg{A}_{i_2} \times \cdots \times \alg{A}_{i_n}$ and any ordered set partition $\pi\in\OSP_n$, we have

\begin{equation}\label{semi-vanishing1}
K_\pi(X_1,X_2, \ldots, X_n)
- \sum_{\tau \in \OSP_n} K_\tau(X_1,X_2,\ldots,X_n)
\, \gold(\tau,\kappa(i_1,\dots,i_n),\pi) = 0.
\end{equation}

We can also formulate the theorem in terms of moments; we only need to replace
(<ref>) by the equation

\begin{equation}
K_\pi(X_1,X_2, \ldots, X_n)
- \sum_{\tau \in \OSP_n}
\phi_\tau(X_1,X_2,\ldots,X_n) \, w(\tau,\kappa(i_1,\dots,i_n),\pi) = 0.
\end{equation}

We fix a tuple $(i_1,\dots, i_n)$ with kernel
It suffices to show that for any tuple of
random variables $(X_1,X_2,\dots,X_n)\in \alg{A}_{i_1} \times \alg{A}_{i_2}
\times \cdots \times \alg{A}_{i_n}$
the following are equivalent:

*  $\phi_\pi(X_1,X_2,\dots,X_n) = \phi_{\pi\curlywedge \eta}(X_1,X_2,\dots,X_n)$ for all $\pi \in \OSP_n$,

*  $K_\pi(X_1,X_2, \ldots, X_n) = \sum_{\tau \in \OSP_n} K_\tau(X_1,X_2,\ldots,X_n) \, \gold(\tau,\eta,\pi)$ for all $\pi \in \OSP_n$.

Assume (<ref>), then

\begin{equation}
\begin{split}
K_\pi(X_1,X_2, \ldots, X_n)
&=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \leq \pi}}
\varphi_\sigma(X_1, X_2,\ldots,X_n)\,\widetilde{\mu}(\sigma,\pi)\\
&=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \leq \pi}}
\varphi_{\sigma\curlywedge\eta}(X_1, X_2,\ldots,X_n)\,\widetilde{\mu}(\sigma,\pi)\\
&=K_\pi(X_1^{(i_1)},X_2^{(i_2)}, \ldots, X_n^{(i_n)}),
\end{split}
\end{equation}

where (<ref>) was used in
the last step. We conclude (<ref>) from Proposition <ref>.

Conversely, assume (<ref>), then Proposition <ref> implies that

\begin{equation}
K_\pi(X_1,X_2, \ldots, X_n)=
K_\pi(X_1^{(i_1)},X_2^{(i_2)}, \ldots, X_n^{(i_n)}).
\end{equation}

On one hand Theorem <ref> implies

\begin{equation}
K_\pi(X_1,X_2, \ldots, X_n)
=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \leq \pi}}
\varphi_\sigma(X_1, X_2,\ldots,X_n)\,\widetilde{\mu}(\sigma,\pi),
\end{equation}

and on the other hand (<ref>) implies

\begin{equation}
K_\pi(X_1^{(i_1)},X_2^{(i_2)}, \ldots, X_n^{(i_n)}) =\sum_{\substack{\sigma \in \OSP_n \\ \sigma \leq \pi}}
\varphi_{\sigma\curlywedge\eta}(X_1, X_2,\ldots,X_n)\,\widetilde{\mu}(\sigma,\pi).
\end{equation}

Thus we get

\begin{equation}
\sum_{\substack{\sigma \in \OSP_n \\ \sigma \leq \pi}}
\varphi_\sigma(X_1, X_2,\ldots,X_n)\,\widetilde{\mu}(\sigma,\pi) = \sum_{\substack{\sigma \in \OSP_n \\ \sigma \leq \pi}}
\varphi_{\sigma\curlywedge\eta}(X_1, X_2,\ldots,X_n)\,\widetilde{\mu}(\sigma,\pi)
\end{equation}

for all $\pi \in \OSP_n$. Finally Möbius inversion yields (<ref>).

It is not obvious that Theorem <ref> generalizes Proposition <ref>, the vanishing of mixed cumulants for $\exch$-independent subalgebras.
In fact it implies a nontrivial identity: for any $ \tau,\eta,\pi \in \OSP_n$ such that $\eta\restr_P \neq \hat{1}_P$ for some $P\in\pi$,

\begin{equation}\label{eq permute}
\sum_{h \in \SG_{|\tau|}} \gold(h(\tau), \eta, \pi)=0,
\end{equation}

where $h(\tau)$ is the action of the permutation $h$ on the blocks of $\tau$.
Similarly, Remark <ref> (or Proposition <ref>)
generalizes the vanishing of mixed cumulants for $\exch$-independent
subalgebras. Consequently we must have

\begin{equation}\label{eq permute2}
\sum_{h \in \SG_{|\tau|}} w(h(\tau), \eta, \pi)=0
\end{equation}

under the same assumptions on $\tau,\eta,\pi.$

Some Goldberg coefficients $\gold(\tau,\eta)$ are known to vanish even when
$\bar{\tau} \leq \bar{\eta}$ (see [65]), while the Weisner coefficients
$w(\tau,\eta)$ do not vanish whenever $\bar{\tau} \leq \bar{\eta}$.
The following Proposition describes two sufficient criteria for vanishing Goldberg
coefficients .

Suppose $\tau,\eta\in\OSP_n$ are such that $\bar{\tau}\leq \bar{\eta}$.

*  If $\des_\eta(\tau)=\asc_\eta(\tau)$ and $\abs{\tau}$ is even then $\gold(\tau,\eta)=0$.

*  If $|\tau|$ is prime then $g(\tau,\eta) \neq0$.

(<ref>) By <cit.>, the coefficient $\gold(\tau,\eta)$ of $K_\tau(X_1,\dots,X_n)$ vanishes if $q_1+\cdots +q_r +\des_\eta(\tau)+ \asc_\eta(\tau)-r$ is odd and $\des_\eta(\tau)=\asc_\eta(\tau)$. Since $\sum_{i=1}^r q_i = r +\plat_\eta(\tau)$ and $\abs{\tau}= \des_\eta(\tau)+\asc_\eta(\tau)+\plat_\eta(\tau)+1$, the conclusion follows.

(<ref>) The idea of the proof is taken from [65]. Let $p:=\abs{\tau}$. By definition and Proposition <ref> we have

\begin{equation}
\begin{split}
\gold(\tau,\eta)
&=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \geq \tau}}
\widetilde{\zeta}(\tau,\sigma)\,w(\sigma,\eta)
=\sum_{\substack{\sigma \in \OSP_n \\ \sigma \geq \tau}}
\frac{1}{[\tau:\sigma]!} \frac{(-1)^{\abs{\sigma}-\asc_\eta(\sigma)-1}}{\abs{\sigma}\binom{\abs{\sigma}-1}{\asc_\eta(\sigma)}} \\
&=  \frac{(-1)^{p-\asc_\eta(\tau)-1}}{p\binom{p-1}{\asc_\eta(\tau)}}
+ \sum_{\substack{\sigma \in \OSP_n \\ \sigma > \tau}}
\frac{1}{[\tau:\sigma]!}
\frac{(-1)^{\abs{\sigma}-\asc_\eta(\sigma)-1}}{\abs{\sigma}\binom{\abs{\sigma}-1}{\asc_\eta(\sigma)}}.
\end{split}
\end{equation}

Since the number $[\tau:\sigma]!
\abs{\sigma}\binom{\abs{\sigma}-1}{\asc_\eta(\sigma)}$ never contains $p$ as a
factor for any $\sigma >\tau$,  $g(\tau,\eta)$ is nonzero.

It is a difficult problem to characterize vanishing Goldberg coefficients.
The criterion  (<ref>) from Proposition <ref> above
does not cover all cases,
for example one can show that $\gold(\tau,\eta)=0$ for
$\tau=(\{3\},\{4\},\{2\},\{1\}), \eta=(\{1,2,3\},\{4\})$,
although the pair $(\tau,\eta)$ does not satisfy the assumption of the criterion.
More information about Goldberg coefficients can be found in
[48, 63]
and in particular  <cit.> concerning the question of vanishing coefficients.

[Goldberg coefficients and partial vanishing of cumulants]
We will write the ordered kernel set partition $\kappa(i_1,\dots, i_n)$ simply
as $i_1i_2\dots i_n$.

* Take $\eta= 12$. We compute $g(12,12)$. Now $\tau=\eta$ so $m_1=1, m_2=2, r=2, q_1=q_2=1$. Hence $\des_\eta(\tau)=0, \asc_\eta(\tau)=1$ and so

\begin{equation}
g(12,12)= \frac{1}{1!1!}\int_{-1}^0 (1+x) \,dx=\frac{1}{2}.
\end{equation}

Similarly  we get $g(21,12)=-\frac{1}{2}, g(12,21)=-\frac{1}{2},
g(21,21)=\frac{1}{2}$ and hence

\begin{align*}
&K_{11}(X^{(1)},Y^{(2)}) = \frac{1}{2}K_{12}(X,Y)-\frac{1}{2} K_{21}(X,Y), \\
& K_{11}(X^{(2)}, Y^{(1)}) =  -\frac{1}{2}K_{12}(X,Y)+\frac{1}{2} K_{21}(X,Y).
\end{align*}

In the most important spreadability systems, like the tensor, free, Boolean or
monotone spreadability systems, partitioned cumulants factorize, e.g.,
Hence we get $K_{11}(X^{(1)},Y^{(2)}) =K_{11}(X^{(2)},Y^{(1)})=0$.

* We then consider the case $\eta=112= (\{1,2\},\{3\})$. The Goldberg coefficients can be nonzero only when $\bar{\tau} \leq \bar{\eta}$, so $\tau$ is one of

112, 221, 123, 132, 213, 231, 312, 321.

If $\tau=112$ then $g(112,112)=\frac{1}{2}$ by the same calculation as $g(12,12)$. If $\tau=221$ then again $g(221,112)= -\frac{1}{2}$.
If $\tau=123$ then $m_1=1,m_2=1,m_3=2, r=2, q_1=2, q_2=1$. So

\begin{equation}
g(123,112) = \frac{1}{1!2!}\int_{-1}^0 (1+x) P_2(x) \, dx = \frac{1}{12}.
\end{equation}

If we take $\tau= 132$ then $m_1=1,m_2=2, m_3=1$, so $r=3,q_1=q_2=q_3=1$, $\asc = 1, \des=1$. Thus we get

\begin{equation}
g(132,112) = \frac{1}{1!1!1!}\int_{-1}^0 x (1+x) \, dx = -\frac{1}{6}.
\end{equation}

If we take $\tau= 231$ then $m_1=2,m_2=1,m_3=1$. So $r=2, q_1=1, q_2=2$, $\asc=0,\des=1$. Hence

\begin{equation}
g(231,112) = \frac{1}{1!2!}\int_{-1}^0 x P_2(x) \, dx = \frac{1}{2}\int_{-1}^0 x (2x+1)\, dx=\frac{1}{12}.
\end{equation}

Similarly we can compute the remaining Goldberg coefficients and get

\begin{equation}
\begin{split}
K_{111}(X^{(1)},Y^{(1)},Z^{(2)}) &= \frac{1}{2} K_{112} -\frac{1}{2} K_{221} + \frac{1}{12} K_{123} -\frac{1}{6} K_{132} \\
&\qquad+\frac{1}{12}K_{213}+\frac{1}{12} K_{231} -\frac{1}{6}K_{312} + \frac{1}{12}K_{321},
\end{split}
\end{equation}

where $X,Y,Z$ are omitted for simplicity. We can see that (<ref>) holds (now $\pi=111$):

\begin{equation}
\frac{1}{2} - \frac{1}{2}=0, \qquad  \frac{1}{12} -  \frac{1}{6} + \frac{1}{12} + \frac{1}{12}  - \frac{1}{6} + \frac{1}{12} =0.
\end{equation}

Again if the cumulants factorize then $K_{112}=K_{221}=K_{11}(X,Y)K_1(Z)$ and
$K_{123}=\cdots= K_{321} = K_1(X)K_1(Y)K_1(Z)$, so the mixed cumulant
$K_{111}(X^{(1)},Y^{(1)},Z^{(2)})$ vanishes.

Similarly one can compute $g(\tau, 121)$ for all $\tau$ such that $\bar{\tau} \leq \overline{121}$ and get

\begin{equation}
\begin{split}
K_{111}(X^{(1)},Y^{(2)},Z^{(1)}) &= \frac{1}{2} K_{121} -\frac{1}{2} K_{212} - \frac{1}{6} K_{123} +\frac{1}{12} K_{132} \\
&\qquad+\frac{1}{12}K_{213}+\frac{1}{12} K_{231} +\frac{1}{12}K_{312} - \frac{1}{6}K_{321}.
\end{split}
\end{equation}

Now in the tensor, free or Boolean spreadability system the mixed cumulant
vanishes. However in the monotone spreadability system it does not:  $K_{212}$
vanishes identically since $212$ is not a monotone partition (see Proposition
<ref>) and therefore, in the monotone case we have

\[
K_{111}(X^{(1)},Y^{(2)},Z^{(1)}) = \frac{1}{2}K_{121}(X,Y,Z)=\frac{1}{2} K_{11}(X,Z)K_1(Y)
\]

which does not vanish in general.

The calculation of these cumulants in terms of moments is easier.

[Weisner coefficients and partial vanishing of cumulants] We can reuse some results from Example <ref>.

* If we take $\tau=\eta=12$ then $\asc_\eta(\tau)=1$ and so

\begin{equation}
w(12,12)= \frac{(-1)^{2-1-1}}{2 \binom{1}{1}}=\frac{1}{2}.
\end{equation}

* Similarly if $\tau=21$ and  $\eta=12$ then $m_1=2, m_1=1$ so $\asc_\eta(\tau)=0$.
Therefore we get $w(21,12)=-\frac{1}{2}$. Similarly, $w(12,21)=-\frac{1}{2}, w(21,21)=\frac{1}{2}$. So

\begin{align*}
&K_{11}(X^{(1)},Y^{(2)}) = \frac{1}{2}\phi_{12}(X,Y)-\frac{1}{2} \phi_{21}(X,Y), \\
& K_{11}(X^{(2)}, Y^{(1)}) =  -\frac{1}{2}\phi_{12}(X,Y)+\frac{1}{2} \phi_{21}(X,Y).
\end{align*}

In factorizing spreadability systems we have
$\phi_{12}(X,Y)=\phi_{21}(X,Y)=\phi_1(X)\,\phi_1(Y)$ and hence $K_{11}(X^{(1)},Y^{(2)}) =K_{11}(X^{(2)},Y^{(1)})=0$.

* One can show that $w(112,112)=\frac{1}{2} = - w(221,112)$ by the same calculation as $w(12,12)$ and $w(21,12)$.
If $\tau=123,\eta=112$ then $m_1=1,m_2=1,m_3=2$, so $\asc =1$ and

\begin{equation}
w(123,112) = \frac{(-1)^{3-1-1}}{3\binom{2}{1}} = -\frac{1}{6}.
\end{equation}

If we take $\tau= 231$ then $m_1=2,m_2=1,m_3=1$ and so $\asc=0$. Hence

\begin{equation}
w(231,112) = \frac{(-1)^{3-0-1}}{3\binom{2}{0}}=\frac{1}{3}.
\end{equation}

Similarly we can compute the remaining Weisner coefficients and get

\begin{equation}
\begin{split}
K_{111}(X^{(1)},Y^{(1)},Z^{(2)}) &= \frac{1}{2} \phi_{112} -\frac{1}{2} \phi_{221} - \frac{1}{6} \phi_{123} -\frac{1}{6} \phi_{132} \\
&\qquad-\frac{1}{6}\phi_{213}+\frac{1}{3} \phi_{231} -\frac{1}{6}\phi_{312} + \frac{1}{3}\phi_{321},
\end{split}
\end{equation}

where $X,Y,Z$ are omitted for simplicity. We can see that (<ref>) holds (now $\pi=111$):

\begin{equation}
\frac{1}{2} - \frac{1}{2}=0, \qquad  -\frac{1}{6} -  \frac{1}{6} - \frac{1}{6} + \frac{1}{3}  - \frac{1}{6} + \frac{1}{3} =0.
\end{equation}

In factorizing spreadability systems the mixed cumulant $K_{111}(X^{(1)},Y^{(1)},Z^{(2)})$ vanishes by using the factorization of partitioned moments.

* Similarly one can compute $w(\tau, 121)$ for all $\tau$ such that $\bar{\tau} \leq \overline{121}$ and get

\begin{equation}
\begin{split}
K_{111}(X^{(1)},Y^{(2)},Z^{(1)}) &= \frac{1}{2} \phi_{121} -\frac{1}{2} \phi_{212} - \frac{1}{6} \phi_{123} -\frac{1}{6} \phi_{132} \\
&\qquad+\frac{1}{3}\phi_{213}-\frac{1}{6} \phi_{231} +\frac{1}{3}\phi_{312} - \frac{1}{6}\phi_{321}.
\end{split}
\end{equation}

In the tensor, free or Boolean spreadability system the mixed cumulant
vanishes, but if it is monotone, then $\phi_{212}=\phi(X)\phi(Y)\phi(Z)$ while
$\phi_{121}=\phi(X Z)\phi(Y)$. Therefore, in the monotone case we have

\[
K_{111}(X^{(1)},Y^{(2)},Z^{(1)}) = \frac{1}{2}(\phi_{121}-\phi_{212})=\frac{1}{2}(\phi(X Z)- \phi(X)\phi(Z)) \phi(Y)
\]

which does not vanish in general.

§ CAMPBELL-BAKER-HAUSDORFF FORMULA AND LIE POLYNOMIALS

The material of the preceding section resembles some results from the theory of
free Lie algebras, cf. the book by C. Reutenauer
[48] already cited above.
In particular,
the  Goldberg coefficients  $\gold(\tau,\eta)$ from
Proposition <ref> coincide with
the coefficients of the Campbell-Baker-Hausdorff series, i.e.,

\log(e^{a_1}e^{a_2}\dotsm e^{a_n}) = \sum_{w: \text{word}} g_w w

when it is expanded in the ring of noncommutative formal power series,
see [21].
Indeed the following paragraphs will provide a new interpretation of the
coefficients of the CBH formula in terms of a certain spreadability system.

§.§ The noncommutative tensor spreadability system $\cS_{\NCtensorT}$

Given an algebra $\alg{A}$
we introduce an operator-valued spreadability system with the following ingredients.

* Put $\varphi=\Id: \mathcal{A}\to \mathcal{A}$.

* $\mathcal{U}:=\otimes_{i=1}^\infty \mathcal{A}$ is the algebraic tensor
product, cf. Section <ref>.

* $\iota^{(j)}: \mathcal{A}\to\mathcal{U}$ is the natural embedding of $\mathcal{A}$ into the $j^{\mathrm{th}}$ component of $\mathcal{U}$:

\[
\iota^{(j)}(X):= 1^{\otimes (j-1)} \otimes X \otimes 1^{\otimes \infty}.
\]

* $\widetilde{\varphi}:=\conc_\infty:  \mathcal{U}\to \mathcal{A}$ is the concatenation product.
This can be alternatively defined by

\widetilde{\varphi}(X_1^{(i_1)} X_2^{(i_2)}\cdots X_n^{(i_n)}):= \phi_\pi(X_1,X_2,\dots,X_n)=X_{P_1} X_{P_2}\cdots X_{P_k},

where $\pi=\kappa(i_1,i_2,\ldots,i_n)=(P_1,P_2,\ldots,P_k)$ and $X_P$ is
defined in (<ref>).

Thus $X_1,X_2,\dots,X_n$ are “shuffled” by $\widetilde{\phi}$ accordingly to the upper indices $(i_1,i_2,\dots,i_n)$. For example

\widetilde{\varphi}(X_1^{(5)}X_2^{(2)}X_3^{(3)}X_4^{(2)}X_5^{(3)}):= X_2 X_4 X_3 X_5 X_1.

We call the triple $\cS_{\NCtensorT}=(\mathcal{U}, (\iota^{(j)})_{j=1}^\infty, \conc_\infty)$ the noncommutative tensor spreadability system.

The above defined triple $\cS_{\NCtensorT}$ is a spreadability system for $(\mathcal{A},\Id)$.

Clearly $\widetilde{\varphi}\circ \iota^{(i)}=\Id$ on
$\mathcal{A}$. The symmetry condition (<ref>) holds too since the value of
$\widetilde{\varphi}$ only depends on the ordered kernel set partition of the upper indices.

The corresponding NCT cumulants $K_\pi^{\NCtensorT}$
satisfy (ordered) multiplicativity like the tensor, free and Boolean cumulants.

For $\pi = (P_1,P_2,\dots, P_p) \in \OSP_n$, we have

\[
K_\pi^{\NCtensorT}(X_1,\dots, X_n) = K_{|P_1|}^{\NCtensorT}(X_{P_1})\,  K_{|P_2|}^{\NCtensorT}(X_{P_2}) \dotsm  K_{|P_p|}^{\NCtensorT}(X_{P_p}).
\]

For $\pi=(P_1,\dots, P_p)\in\OSP_n$,

\begin{equation}
\begin{split}
\varphi_\pi(N.X_1, \dots, N.X_n)
&= \sum_{i_1, \dots, i_n \in [N]} \varphi_{\pi\curlywedge \kappa(i_1,\dots, i_n)}(X_1, \dots, X_n) \\
&= \sum_{i_1, \dots, i_n \in [N]}  X_{\kappa(i_1,\dots, i_n) \restr_{P_1}} X_{\kappa(i_1,\dots, i_n) \restr_{P_2}} \cdots X_{\kappa(i_1,\dots, i_n) \restr_{P_p}}\\
&= \sum_{\substack{i_k \in [N] \\ k\in P_1}}  X_{\kappa(i_1,\dots, i_n) \restr_{P_1}}  \sum_{\substack{i_k \in [N] \\ k\in P_2}}  X_{\kappa(i_1,\dots, i_n) \restr_{P_2}} \cdots  \sum_{\substack{i_k \in [N] \\ k\in P_p}} X_{\kappa(i_1,\dots, i_n) \restr_{P_p}}\\
&= \widetilde{\varphi}\!\left(\prod_{i\in P_1}N.X_i\right)\cdots \widetilde{\varphi}\!\left(\prod_{i\in P_p}N.X_i\right).
\end{split}
\end{equation}

The conclusion follows by comparing the coefficients of $N^{p}$.

After having established multiplicativity it suffices to compute
Theorem <ref> reads as follows.

For $n \in \N$ we have

\[
K_n^{\NCtensorT} (X_1,\dots, X_n) = \sum_{\pi =(P_1,P_2,\dots)\in \OSP_n} \frac{(-1)^{|\pi|-1}}{|\pi|}X_{P_1} X_{P_2}\cdots X_{P_{|\pi|}}.
\]

The reader can easily verify that

\begin{align}
K_1^{\NCtensorT}(X)&=X, \\
K_2^{\NCtensorT}(X_1,X_2) &= \frac{1}{2}[X_1,X_2], \\
K_3^{\NCtensorT}(X_1,X_2,X_3) &= \frac{1}{3} (X_1 X_2 X_3+X_3 X_2 X_1) \\
&\qquad- \frac{1}{6} (X_1 X_3 X_2 + X_2 X_1 X_3 + X_2 X_3 X_1 + X_3 X_1 X_2).   \notag
\end{align}

It can be shown that $K_n^{\NCtensorT}~(n\geq2)$ can be expressed
as a sum of commutators. For the proof it suffices to show that
$K_n^{\freeLie}$ defined later can be written as a sum of commutators, which follows from the fact that $K_n^{\freeLie}(a_1,\dots,a_n)$ is a Lie polynomial (see Remark <ref>) and every Lie polynomial is a fixed point of the (linear extension of) map $a_1a_2\cdots a_n \mapsto n^{-1}[a_1,[a_2,[\cdots,[a_{n-1},a_n]] \cdots]$ (see  <cit.>).

The CBH formula on $\alg{A}$ can be expressed in terms of NCT cumulants.

As formal power series on $\mathcal{A}$ we have the identity

\[
\begin{split}
&\log(e^{a_1} e^{a_2}\cdots e^{a_n}) \\
&\qquad= \sum_{\substack{(p_1,p_2,\dots,p_n) \in (\N\cup \{0\})^n,\\(p_1,p_2,\dots,p_n) \neq (0,0,\dots,0)  }} \frac{1}{p_1!p_2! \cdots p_n!}K^{\NCtensorT}_{p_1+p_2+\cdots+p_n}(\underbrace{a_{1}, a_{1},\ldots, a_{1}}_{\text{$p_1$ times}},\ldots, \underbrace{a_{n},a_n,\ldots,a_{n}}_{\text{$p_n$  times}}).
\end{split}
\]

First observe that

\begin{equation}\label{log1}
\begin{split}
&1+ \sum_{\substack{(p_1,p_2,\dots,p_n) \in (\N\cup \{0\})^n,\\(p_1,p_2,\dots,p_n) \neq (0,0,\dots,0)  }} \frac{1}{p_1!p_2! \cdots p_n!}\widetilde{\varphi}((N.a_{1})^{p_1}(N.a_2)^{p_2}\cdots(N.a_{n})^{p_n})\\
&\qquad= \widetilde{\varphi}(e^{N.a_1} \cdots e^{N.a_n}) \\
&\qquad =  \widetilde{\varphi}(e^{a_1^{(1)}+\cdots + a_1^{(N)}} \cdots e^{a_n^{(1)}+\cdots + a_n^{(N)}}).
\end{split}
\end{equation}

Our construction implies that $a_i^{(j)}, j=1,2,3,\dots$ commute for each
fixed $i$, so

\begin{equation}
\begin{split}
&\widetilde{\varphi}(e^{a_1^{(1)}+\cdots + a_1^{(N)}} \cdots e^{a_n^{(1)}+\cdots + a_n^{(N)}}) \\
& \qquad=   \widetilde{\varphi}((e^{a_1^{(1)}}e^{a_1^{(2)}}\cdots e^{a_1^{(N)}})( e^{a_2^{(1)}} e^{a_2^{(2)}}\cdots e^{a_2^{(N)}}) \cdots ( e^{a_n^{(1)}} e^{a_n^{(2)}}\cdots e^{a_n^{(N)}}) ).
\end{split}
\end{equation}

By the definition of $\widetilde{\varphi}$ the last expression equals

\begin{equation}\label{log2}
(e^{a_1} e^{a_2}\cdots e^{a_n})^N.
\end{equation}

We conclude by comparing the coefficient of $N$ in the identity

\begin{equation}\label{log3}
\begin{aligned}
e^{N\log(e^{a_1} e^{a_2}\cdots e^{a_n})}
&=(e^{a_1} e^{a_2}\cdots e^{a_n})^N \\
&=1+ \sum_{\substack{(p_1,p_2,\dots,p_n) \in (\N\cup \{0\})^n,\\(p_1,p_2,\dots,p_n) \neq (0,0,\dots,0)  }} \frac{1}{p_1!p_2! \cdots p_n!}\widetilde{\varphi}((N.a_{1})^{p_1}(N.a_2)^{p_2}\cdots(N.a_{n})^{p_n}).
\end{aligned}
\end{equation}

Theorem <ref> is similar to the well-known formula in probability theory,

\begin{equation}
\begin{split}
&\log \E[e^{z_1 X_1 +\cdots +z_n X_n}] \\
&\quad=  \sum_{\substack{(p_1,p_2,\dots,p_n) \in (\N\cup \{0\})^n,\\(p_1,p_2,\dots,p_n) \neq (0,0,\dots,0)  }} \frac{z_1^{p_1}z_2^{p_2}\cdots z_n^{p_n}}{p_1!p_2! \cdots p_n!}K^{\tensorT}_{p_1+p_2+\cdots+p_n}(\underbrace{X_{1}, X_{1},\ldots, X_{1}}_{\text{$p_1$  times}},\ldots, \underbrace{X_{n},X_n,\ldots,X_{n}}_{\text{$p_n$ times}}),
\end{split}
\end{equation}

where $X_1,\dots,X_n$ are $\C$-valued classical random variables and $z_1,\dots, z_n$ are commutative indeterminates.

*  A sequence $(\mathcal{A}_i)_{i=1}^\infty$ of subalgebras of $\mathcal{A}$ is $\cS_{\NCtensorT}$-independent if and only if the subalgebras $\mathcal{A}_1, \mathcal{A}_2, \dots$ commute mutually. This is obviously equivalent to $K_2^{\NCtensorT}(X,Y)=0$ whenever $X \in \mathcal{A}_i, Y\in \mathcal{A}_j$ with $i\neq j$.

*  For fixed $n \geq2$, if $\{X_1,\dots, X_n\} \subseteq \mathcal{A}$ splits into two mutually commutative families then $K_n^{\NCtensorT}(X_1,\dots, X_n)=0$.

* Additivity of Lie polynomials in commuting variables is well known,
see <cit.>).

* We have thus shown that

\[
\text{$\cS_{\NCtensorT}$-independence = commutativity = vanishing of mixed NCT cumulants}.
\]

This example illustrates that in general vanishing of mixed cumulants does not imply exchangeability.

(<ref>)   Suppose that $(\mathcal{A}_i)_{i=1}^\infty$ is $\cS_{\NCtensorT}$-independent. Let $\pi =\hat{1}_2, \rho = (\{2\},\{1\}) \in\OSP_2$ and let $X \in \mathcal{A}_i, Y \in \mathcal{A}_j$ for fixed $i > j$. Then $\kappa(i,j)=\rho$ and

\begin{equation}
\varphi_\pi(X,Y) = X Y, \qquad \varphi_{\pi\curlywedge \rho}(X,Y) = Y X,
\end{equation}

and by $\cS_{\NCtensorT}$-independence these two must coincide, so $X Y = Y X$. This shows that $[\mathcal{A}_i,\mathcal{A}_j]=0$.  Conversely, if the subalgebras $\mathcal{A}_1, \mathcal{A}_2, \dots$ are mutually commutative then for any $i_1,\dots, i_n \in \N$, any $X_k \in \mathcal{A}_{i_k}, k=1,\dots, n$ and any $\pi=(P_1,\dots, P_p) \in\OSP_n$, let $\rho:=\kappa(i_1,\dots, i_n)=(R_1,\dots, R_r)$. Then

\begin{equation}
\varphi_{\pi\curlywedge \rho}(X_1,\dots, X_n) = (X_{P_1 \cap R_1} X_{P_1 \cap R_2} \cdots X_{P_1 \cap R_r} )  \cdots (X_{P_p \cap R_1} X_{P_p \cap R_2} \cdots X_{P_p \cap R_r})
\end{equation}

where $X_\emptyset$ is understood as the unit $\hat{1}_\mathcal{A}$.
By commutativity, $\{X_k \mid k \in R_i\}$ and $\{X_k \mid k \in R_j\}$ commute
for distinct $i,j$ and consequently for each $i =1,\dots, p$ we have

\begin{equation}
X_{P_i \cap R_1} X_{P_i \cap R_2} \cdots X_{P_i \cap R_r} = X_{P_i},
\end{equation}

which shows that $\varphi_{\pi\curlywedge \rho}(X_1,\dots, X_n) =\varphi_{\pi}(X_1,\dots, X_n) $.

(<ref>)   Suppose that $\{X_i \mid i\in I\}$ and $\{X_i \mid i\in I^c\}$ commute with
each other and $\emptyset \subsetneq I \subsetneq \{1,\dots,n\}$. Then, for
commutative indeterminates $z_1,\dots, z_n$, we have

\begin{equation}\label{sum log}
\log(e^{z_1 X_1} \cdots e^{z_n X_n}) = \log\left(\prod_{i \in I} e^{z_ i X_i}\right) + \log\left(\prod_{i \in I^c} e^{z_ iX_i}\right),
\end{equation}

where the products $\prod_{i \in I}, \prod_{i \in I^c}$ preserve the natural orders on $I,I^c$.   On the other hand, by Theorem <ref>, $ \log(e^{z_1 X_1} \cdots e^{z_n X_n})$ equals

\begin{equation}\label{CBH d}
\sum_{\substack{(p_1,p_2,\dots,p_n) \in (\N\cup \{0\})^n,\\(p_1,p_2,\dots,p_n) \neq (0,0,\dots,0)  }} \frac{z_1^{p_1} z_2^{p_2} \cdots z_n^{p_n}}{p_1!p_2! \cdots p_n!}K^{\NCtensorT}_{p_1+p_2+\cdots+p_n}(\underbrace{X_{1}, X_{1},\ldots, X_{1}}_{\text{$p_1$ times}},\ldots, \underbrace{X_{n},X_n,\ldots,X_{n}}_{\text{$p_n$ times}}).
\end{equation}

If we compare the coefficients of $z_1z_2\cdots z_n$ in (<ref>) and (<ref>) then we get $K_n(X_1,\dots, X_n)=0$.

§.§ Specialization to free algebras

We restrict the noncommutative tensor spreadability system $\cS_{\NCtensorT}$ to the case when the underlying algebra $\mathcal{A}$ is a free algebra.
The aim of this section is to show that the NCT cumulants are Lie polynomials
in the following sense (cf. [48]).  Let $\fA$ be an alphabet, whose elements are typically denoted by
$a_1,a_2,\dots$, and let $\fA^\ast$ be the free monoid generated
by $\fA$ endowed with the concatenation product. An element of
$\fA^\ast$ is called a word. If $w=a_1 a_2\cdots a_n$,
$a_i\in\fA$, then the word $w$ has length $n$. The length of
the unit $1$ is understood to be 0. Let $\C[\fA]$
be the monoid algebra of the monoid $\fA^\ast$ over $\IC$.

We construct the noncommutative tensor spreadability system for $(\C[\fA], \text{Id})$ as in Section <ref>. In this case we call it free Lie spreadability system and denote it by $\cS_{\freeLie}$ instead of $\cS_{\NCtensorT}$.

Moreover, on free algebras we can introduce a bialgebra structure.  Let $\delta_k: \C[\fA]\to\C[\fA]^{\otimes k}$ be the
coproduct uniquely determined by the values

\delta_k(a)= \sum_{n=0}^{k-1} 1^{\otimes n} \otimes a \otimes 1^{\otimes (k-n-1)},\qquad a\in\fA

on the generators (compare Definition <ref>).
In the language of Hopf algebras, this is the $k$-fold
copower of the unique coproduct $\delta$ for which every generator is a
primitive element.
Moreover, in this case the dot operation  corresponds to the coproduct, that is,

\[
N.a = \delta_N(a) \otimes 1^{\otimes \infty}, \qquad a \in \fA.
\]

The counit is the unique linear map $\epsilon:\C[\fA] \to \C$ such that

\epsilon(1)=1,\qquad
\epsilon(w)=0, \quad w\in\fA^\ast \setminus\{1\}.

Let $\conc_k: \C[\fA]^{\otimes k} \to \C[\fA]$ be the linear map defined by

\conc_k(X_1\otimes X_2\otimes\cdots \otimes X_k)=X_1X_2\cdots X_k,\qquad X_i\in\C[\fA].

For endomorphisms $f_1, f_2,\dots, f_k$ of $\C[\fA]$, we define the convolution

f_1 \ast f_2\ast\cdots \ast f_k :=\conc_k \circ (f_1\otimes f_2\otimes\cdots \otimes f_k) \circ \delta_k.

Let $\Pi: \C[\fA] \to \C[\fA]$ be the map defined by

\Pi=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} (\Id-\epsilon)^{\ast k},

where $\epsilon$ is regarded as an endomorphism of $\C[\fA]$.
It follows from Proposition <ref> below that
for any $a_i \in\fA$ the value $\Pi(a_1 a_2\cdots a_n)$ is a
finite sum of words and that indeed  $\Pi(\C[\fA] ) \subseteq
\C[\fA]$.

If $a_1,a_2,\dots,a_n \in \fA$ and $k\in\N$, then

\delta_k(a_1a_2\cdots a_n)= \sum_{\substack{\pi =(P_1,P_2,\dots) \in \EOSP_n\\
\abs{\pi}=k}} a_{P_1}\otimes a_{P_2}\otimes \cdots \otimes a_{P_k},

where the sum runs over pseudopartitions, see Definition <ref>.

Since  $\delta_k$ is a homomorphism, we have

\begin{equation}
\begin{split}
\delta_k(a_1a_2\cdots a_n)
&=  \delta_k(a_1)\, \delta_k(a_2) \cdots \delta_k(a_n)\\
&= \left(a_1 \otimes 1 \otimes \cdots \otimes 1
1 \otimes a_1 \otimes \cdots \otimes 1
\cdots
1 \otimes  \cdots \otimes 1\otimes a_1 \right)   \\
\left(a_2 \otimes 1 \otimes \cdots \otimes 1
1 \otimes a_2 \otimes \cdots \otimes 1
\cdots
1 \otimes  \cdots \otimes 1\otimes a_2 \right)   \\
&\quad \cdots\left(
a_n \otimes 1 \otimes \cdots \otimes 1
1 \otimes  a_n \otimes \cdots \otimes 1
\cdots
+ 1 \otimes \cdots \otimes 1\otimes a_n \right),
\end{split}
\end{equation}

which is equal to the desired expression in terms of ordered pseudopartitions.

Now we have the following expression for $\Pi$.

Let $a_1,a_2,\ldots,a_n \in \fA$. Then

\Pi(a_1a_2\cdots a_n) = \sum_{\pi = (P_1, P_2, \dots) \in \OSP_n}
\frac{(-1)^{\abs{\pi}-1}}{\abs{\pi}} a_{P_1} a_{P_2}\cdots a_{P_{\abs{\pi}}}.

From Proposition <ref> we have

\begin{equation*}
\begin{split}
\Pi(a_1a_2\cdots a_n)
&= \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} (\Id-\epsilon)^{\ast k}(a_1a_2\cdots a_n)\\
&=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} (\conc_k \circ (\Id-\epsilon)^{\otimes k}\circ \delta_k)(a_1a_2\cdots a_n) \\
&=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{\substack{\pi =(P_1,P_2,\dots)
\in \EOSP_n\\ \abs{\pi}=k}} (\Id-\epsilon)(a_{P_1}) (\Id-\epsilon)(a_{P_2}) \cdots (\Id-\epsilon)(a_{P_k}).
\end{split}
\end{equation*}

Note that $(\Id-\epsilon)(1)=0$ and $(\Id-\epsilon)(w)=w$ for
$w\in\fA^\ast\setminus\{1\}$. If $P_i=\emptyset$ for some $i$, then
the product $(\Id-\epsilon)(a_{P_1}) (\Id-\epsilon)(a_{P_2}) \cdots (\Id-\epsilon)(a_{P_k})$ vanishes. Therefore only proper ordered set partitions contribute to the sum, and in particular $k \leq n$.  This proves the claim.

Combining Proposition <ref> with Proposition <ref> we
conclude the following identity.

Let $a_1,a_2,\ldots,a_n \in \fA$ and let $K^{\freeLie}_\pi$ be the cumulants associated to the spreadability system $\cS_{\freeLie}$. Then

\Pi(a_1a_2\cdots a_n) = K^{\freeLie}_n(a_1,a_2,\dots,a_n).

* The set of Lie polynomials $\Lie(\fA)$ is the smallest subspace
of $\C[\fA]$ that contains $\fA$ and is closed with respect to the Lie bracket
$[X,Y]=XY-YX$. It is well known that $\Pi$ is a Lie projector, i.e.,
$\Lie(\fA)=\Pi(\C[\fA])$ <cit.>.
So in the above setting cumulants with entries from $\fA$ are exactly Lie

* The values at words of higher order are given by convolution
powers of $\Pi$, i.e.,

\begin{equation}
\Pi_k=\frac{1}{k!}\Pi^{\ast k},\qquad k\in\N,
\end{equation}

(see [48]),
and we can show that

\begin{equation}
\Pi_k(a_1a_2\cdots a_n) =\frac{1}{k!}\sum_{\substack{\pi\in \OSP_n\\\abs{\pi}=k}} K^{\freeLie}_\pi(a_1,a_2,\dots,a_n),\qquad a_i\in\fA,~ i\in[n].
\end{equation}

is the coefficient of $N^{k}$ appearing in $\widetilde{\phi}((N.a_1) (N.a_2)\cdots (N.a_n))$ by Theorem <ref> and Theorem <ref>. In other words,

\begin{equation}
\widetilde{\phi}((N.a_1) (N.a_2)\cdots (N.a_n))= \sum_{k=1}^n N^k \Pi_k(a_1a_2\cdots a_n),\qquad a_i\in\fA,~ i\in[n].
\end{equation}

Now the combination of Theorem <ref> and Theorem <ref>
provides yet another proof of the  CBH formula

\begin{equation}\label{CBH}
\log(e^{a_1} e^{a_2}\cdots e^{a_n}) = \sum_{\substack{(p_1,p_2,\dots,p_n) \in
(\N\cup \{0\})^n,\\(p_1,p_2,\dots,p_n) \neq (0,0,\dots,0)  }} \Pi \left(
\frac{a_1^{p_1} a_2^{p_2}\cdots a_n^{p_n}}{p_1! p_2!\cdots p_n!}\right);
\end{equation}

see <cit.>.

§.§ Coefficients of the Campbell-Baker-Hausdorff formula

We adopt the notations and definitions in the previous subsection.
The coefficients of the Campbell-Baker-Hausdorff formula, when written out in
the monomial basis, were first computed using generating functions in
[21] and are
called Goldberg coefficients; a combinatorial proof
can be found in <cit.>.
In the following we give another derivation of the Goldberg coefficients;
see also a recent proof using the theory of noncommutative symmetric functions [20].

Let $\eta=(E_1,E_2,\dots,E_e)$ be an interval partition of $[n]$ where the blocks are in canonical order, i.e. if $s<t$ then $i<j$ for all $ i\in E_s, j\in E_t$.
Then for any $X_i \in \C[\fA]$ and any $\pi\in \OSP_n$, we have

\varphi_{\pi \curlywedge \eta}(X_1,X_2,\dots,X_n)=\varphi_{\pi}(X_1,X_2,\dots,X_n).

The statement holds by definition because every block $P\in\pi$ is the concatenation
of $P\cap E_1, P\cap E_2, \dots, P\cap E_e$.

Thus a sequence of distinct letters satisfies some partial independence, but it is not $\cS_{\freeLie}$-independent. Indeed, $(a_1,a_2)\in \fA \times \fA$ is $\cS_{\freeLie}$-independent if and only if $a_1=a_2$ by Proposition <ref>.

Let $a_1,a_2,\ldots,a_n$ be distinct letters from the alphabet  $\fA$, let $r\in\N$ and
$q_j,i_j \in \N$ for $j\in [m]$ such that $i_j \neq i_{j+1}$ for $j\in[m-1]$.
Then the coefficient of the monomial $a_{i_1}^{q_1}a_{i_2}^{q_2}\cdots
a_{i_m}^{q_m}$ appearing in $\log(e^{a_1}\cdots e^{a_n})$ is given by

\begin{equation}\label{eq:CBH coefficient}
\frac{1}{q_1!q_2!\cdots q_m!}\int_{-1}^0 x^{\des(\underline{i})} (1+x)^{\asc(\underline{i})} \prod_{j=1}^m P_{q_j}(x)\,dx,
\end{equation}

where $P_q(x)$ are the homogeneous Euler polynomials already encountered in Proposition <ref>.

In order for a monomial $v=a_{i_1}^{q_1}a_{i_2}^{q_2}\cdots a_{i_m}^{q_r}$ to
occur as a term in $\Pi(a_1^{p_1}a_2^{p_2}\cdots a_n^{p_n})$ it necessarily
has to be a rearrangement of the word (= permutation of the multiset)
$w=a_1^{p_1}a_2^{p_2}\dotsm a_n^{p_n}$,
since the projector in the CBH formula (<ref>) does not
change multiplicities and therefore
every letter must occur the same number of times in $v$ and $w$.

Thus $p_k=\sum_{j\in B_k}q_j$ where  $B_k=\{j : i_j=k\}$.
Let $p=p_1+p_2+\dots+p_n$ be the total length of $w$
and $\eta=(A_1,A_2,\dots,A_n)\in \OIP_p$ be the ordered interval partition
corresponding to the composition $(p_1,p_2,\dots,p_n)$, i.e.,
and $\abs{A_j}=p_j$.

From Theorem <ref> we infer

\Pi(a_{1}^{p_1}a_{2}^{p_2} \cdots a_{n}^{p_n}) =
K^{\freeLie}_{p}(\underbrace{a_1, a_1,\ldots, a_1}_{\text{$p_1$ times}},
\underbrace{a_2, a_2,\ldots, a_2}_{\text{$p_2$ times}},
\ldots,
\underbrace{a_{n},a_{n},\ldots,a_{n}}_{\text{$p_n$ times}})
=:K_p^{\freeLie}(a_1^{\dots p_1},a_2^{\dots p_2}, \dots, a_n^{\dots p_n}).

Note that by Lemma <ref> we have
$\varphi_{\pi}(a_1^{\dots p_1},a_2^{\dots p_2}, \dots, a_n^{\dots p_n})
= \varphi_{\pi \curlywedge \eta}(a_1^{\dots p_1},a_2^{\dots p_2}, \dots,a_n^{\dots p_n})$
for any $\pi \in \OSP_p$.

\begin{equation}
\begin{split}
K^{\freeLie}_{p}(a_1^{\dots p_1},a_2^{\dots p_2}, \dots, a_n^{\dots p_n})
&= \sum_{\sigma \in \OSP_p} \varphi_\sigma(a_1^{\dots p_1},a_2^{\dots p_2}, \dots, a_n^{\dots p_n})
\, \widetilde{\mu}(\sigma, \hat{1}_p)
\\
&= \sum_{\sigma \in \OSP_p} \varphi_{\sigma\curlywedge\eta}
(a_1^{\dots p_1},a_2^{\dots p_2}, \dots, a_n^{\dots p_n})
\, \widetilde{\mu}(\sigma, \hat{1}_p)\\
&= \sum_{\substack{\tau \in \OSP_p \\ \bar{\tau}\leq \bar{\eta}}}
\varphi_{\tau}(a_1^{\dots p_1},a_2^{\dots p_2}, \dots, a_n^{\dots p_n})
\,w(\tau, \eta)
\end{split}
\end{equation}

by Proposition <ref>.
Thus in order to determine the coefficient of $v$ we must
collect all ordered set partitions $\tau$ such that

\begin{equation}\label{desire}
T_v =\{\tau\in\OSP_p : \bar{\tau}\leq\bar{\eta},
\varphi_{\tau}(a_1^{\dots p_1},a_2^{\dots p_2}, \dots, a_n^{\dots p_n}) = v
\}
\end{equation}

and then sum up the corresponding values of $w(\tau,\eta)$.
Let us now investigate the structure of this set.
First note that $T_v$ is an order ideal: if $\tau\in T_v$ and $\tau'\leq \tau$,
then $\tau'\in T_v$, because every block of $\tau$ contains only repetitions
of one letter and further refinement of $\tau$ leaves the end result
$v$ invariant.
Moreover $T_v$ is the disjoint union of the principal ideals
$\downset\tau_0=\{\tau : \tau\leq \tau_0\}$
where $\tau_0\in T_v$ is maximal.

The maximal partition $\tau_0$ arises as follows:
After application of $\phi_\tau$ for $\tau\in T_v$
each factor $a_j^{p_j}$ is divided into pieces
$a_j^{q_l}$, $l\in B_j$ and the number of such subdivisions is
the multinomial coefficient

\binom{p_j}{q_l : l\in B_j}

These subdivisions are in one-to-one correspondence with
ordered set partitions of $A_j$ which, pieced together in the order of $j$,
give rise to a maximal ordered set partition $\tau_0\in T_v$.
Thus in total there are  as many maximal ordered set partitions
as there are subdivisions, namely

\begin{equation}\label{multinomial}
\binom{p_1}{q_l:l\in B_1}
\binom{p_2}{q_l:l\in B_2}
\dotsm
\binom{p_n}{q_l:l\in B_n}
\frac{p_1!p_2!\cdots p_n!}{q_1!q_2!\cdots q_m!}
\end{equation}

By Proposition <ref> the principal
ideal $\downset\tau_0$ generated by a maximal ordered set partition $\tau_0$
is isomorphic to

\OSP_{q_1}\times\OSP_{q_2}\times\dotsm\OSP_{q_m}

and in particular, all principal ideals are isomorphic.
Moreover the number of ascents
does not depend on the choice of $\tau\in T_v$, indeed it is equal to
Thus all ideals $\downset\tau_0$ deliver the same contribution
and as a consequence of the discussion above we are left with

\sum_{\tau\in T_v} w(\tau,\eta)
\frac{p_1!p_2!\cdots p_n!}{q_1!q_2!\cdots q_m!}
\sum_{\tau\leq\tau_0} w(\tau,\eta)

for one fixed maximal element $\tau_0\in T_v$.

A canonical representative $\tau_0$ is obtained by concatenating
consecutive subintervals of length $q_j$ from $A_{i_j}$, $j=1,2,\dots,m$.
Let us now turn to the value of $w(\tau,\eta)$.
As seen above, the numbers of both ascents and descents of $\tau$
only depend on those of the sequence $\underline{i}$
and since $\underline{i}$ has no plateaux we infer from (<ref>)
that $m=\des(\underline{i}) + \asc(\underline{i})+1$.
On the other hand, if we denote by $(\tau_1,\tau_2,\dots,\tau_m)$
the image of $\tau\in\downset\tau_0$
under the isomorphism of Proposition <ref>
then the first exponent in formula    (<ref>)  of Proposition <ref> becomes

\abs{\tau}-\asc_\eta(\tau) - 1
= \sum_{j=1}^m \abs{\tau_j}-\asc(\underline{i}) - 1
= \sum_{j=1}^m (\abs{\tau_j}-1) + \des(\underline{i})

and thus

w(\tau,\eta) = \int_{-1}^0 x^{\des(\underline{i})}(1+x)^{\asc(\underline{i})} \prod_{j=1}^m x^{\abs{\tau_j}-1} \, d x

and summing over the cartesian product yields the total value

\sum_{\tau\in T_v} w(\tau,\eta)
\frac{p_1!p_2!\cdots p_n!}{q_1!q_2!\cdots q_m!}
\int_{-1}^0 x^{\des(\underline{i})}(1+x)^{\asc(\underline{i})} \prod_{j=1}^m
\left(
\sum_{\rho\in\OSP_{q_j}}
\right)  d x

Finally note that

\begin{equation}
\sum_{\rho\in\OSP_{q}}x^{\abs{\rho}-1}= \sum_{\sigma \in \SP_{q}} \abs{\sigma}!\, x^{\abs{\sigma}-1} =\sum_{k=1}^{q} k!\, S(q,k)  x^{k-1}.
\end{equation}

is indeed the homogeneous Euler polynomial as claimed,
see Remark <ref>.

The authors were not able to prove Theorem <ref> as a corollary of Proposition <ref>, although Goldberg coefficients appear in both formulas.

§ OPEN PROBLEMS

* Find a proof of Theorem <ref> based on Proposition <ref>, or vice versa.

* Is there an application of spreadability systems to noncommutative quasi-symmetric functions (see [12]) or vice versa?

§ ACKNOWLEDGEMENT
The first-named author was supported by European Commission, Marie Curie
Actions IIF 328112 ICNCP and by JSPS Grant-in-Aid for Young Scientists (B) 15K17549. We thank Hayato Saigo for discussions in the early stage of this work and J.-C. Novelli and J.Y. Thibon for pointing out errors in an earlier version.

[1]
Luigi Accardi, Yukihiro Hashimoto, and Nobuaki Obata, A role of
singletons in quantum central limit theorems, J. Korean Math. Soc.
35 (1998), no. 3, 675–690, International Conference on Probability
Theory and its Applications (Taejon, 1998).

[2]
Michael Anshelevich, Partition-dependent stochastic measures and
$q$-deformed cumulants, Doc. Math. 6 (2001), 343–384.

[3]
Daniel Avitzour, Free products of $C^{\ast} $-algebras, Trans. Amer.
Math. Soc. 271 (1982), no. 2, 423–435.

[4]
M. Barnabei, A. Brini, and Dzh.-K. Rota, The theory of Möbius
functions, Uspekhi Mat. Nauk 41 (1986), no. 3(249), 113–157,
Translated from the Italian by S. I. Gel'fand.

[5]
Marek Bożejko, Michael Leinert, and Roland Speicher, Convolution and
limit theorems for conditionally free random variables, Pacific J. Math.
175 (1996), no. 2, 357–388.

[6]
Miklós Bóna, Combinatorics of permutations, second ed., Discrete
Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL,

[7]
Marek Bożejko, Positive definite functions on the free group and the
noncommutative Riesz product, Boll. Un. Mat. Ital. A (6) 5
(1986), no. 1, 13–21.

[8]
Francesco Brenti, Permutation enumeration symmetric functions, and
unimodality, Pacific J. Math. 157 (1993), no. 1, 1–28.

[9]
Kenneth S. Brown, Semigroups, rings, and Markov chains, J. Theoret.
Probab. 13 (2000), no. 3, 871–938.

[10]
, Semigroup and ring theoretical methods in probability,
Representations of finite dimensional algebras and related topics in Lie
theory and geometry, Fields Inst. Commun., vol. 40, Amer. Math. Soc.,
Providence, RI, 2004, pp. 3–26.

[11]
Louis J. Billera and A. Sarangarajan, The combinatorics of permutation
polytopes, Formal power series and algebraic combinatorics (New
Brunswick, NJ, 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci.,
vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 1–23.

[12]
Nantel Bergeron and Mike Zabrocki, The Hopf algebras of symmetric
functions and quasi-symmetric functions in non-commutative variables are free
and co-free, J. Algebra Appl. 8 (2009), no. 4, 581–600.

[13]
Louis Comtet, Advanced combinatorics, enlarged ed., D. Reidel Publishing
Co., Dordrecht, 1974, The art of finite and infinite expansions.

[14]
Adrian Dacko, V-monotone independence, arXiv:1901.06342, 2019.

[15]
J. F. Dillon and D. P. Roselle, Simon Newcomb's problem, SIAM J. Appl.
Math. 17 (1969), 1086–1093.

[16]
Peter Doubilet, Gian-Carlo Rota, and Richard Stanley, On the foundations
of combinatorial theory. VI. The idea of generating function,
Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and
Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II:
Probability theory (Berkeley, Calif.), Univ. California Press, 1972,
pp. 267–318.

[17]
Kurusch Ebrahimi-Fard and Frédéric Patras, The splitting process
in free probability theory, Int. Math. Res. Not. IMRN (2016), no. 9,

[18]
Kurusch Ebrahimi-Fard and Frédéric Patras, Monotone, free, and
boolean cumulants: a shuffle algebra approach, Adv. Math. 328
(2018), 112–132.

[19]
Roland Friedrich and John McKay, Homogeneous Lie groups and quantum
probability, arXiv:1506.07089, 2015.

[20]
Loï c Foissy, Frédéric Patras, and Jean-Yves Thibon, Deformations
of shuffles and quasi-shuffles, Ann. Inst. Fourier (Grenoble) 66
(2016), no. 1, 209–237.

[21]
Karl Goldberg, The formal power series for ${\rm log}\,e^xe^y$, Duke
Math. J. 23 (1956), 13–21.

[22]
I. J. Good, A new formula for cumulants, Math. Proc. Cambridge Philos.
Soc. 78 (1975), no. 2, 333–337.

[23]
Takahiro Hasebe, New associative product of three states generalizing
free, monotone, anti-monotone, boolean, conditionally free and conditionally
monotone products, arXiv:1009.1505, 2010.

[24]
Takahiro Hasebe, Conditionally monotone independence I: Independence,
additive convolutions and related convolutions, Infin. Dimens. Anal. Quantum
Probab. Relat. Top. 14 (2011), no. 3, 465–516.

[25]
Takahiro Hasebe and Hayato Saigo, Joint cumulants for natural
independence, Electron. Commun. Probab. 16 (2011), 491–506.

[26]
, The monotone cumulants, Ann. Inst. Henri Poincaré Probab.
Stat. 47 (2011), no. 4, 1160–1170.

[27]
, On operator-valued monotone independence, Nagoya Math. J.
215 (2014), 151–167.

[28]
André Joyal, Une théorie combinatoire des séries formelles, Adv.
in Math. 42 (1981), no. 1, 1–82.

[29]
Daniel Krob, Matthieu Latapy, Jean-Christophe Novelli, Ha Duong Phan, and
Sylviane Schwer, Pseudo-permutations I:first combinatorial and
lattice properties, FPSAC'01 (H. Barcelo, ed.), 2000.

[30]
Claus Köstler, A noncommutative extended de Finetti theorem, J.
Funct. Anal. 258 (2010), 1073–1120.

[31]
Franz Lehner, Cumulants in noncommutative probability theory I.
Noncommutative exchangeability systems, Math. Zeitschr. 248
(2004), no. 1, 67–100, arXiv:math.CO/0210442.

[32]
Franz Lehner, Cumulants in noncommutative probability theory III.
Creation and annihilation operators on Fock spaces, Infin. Dimens. Anal.
Quantum Probab. Relat. Top. 8 (2005), no. 3, 407–437.

[33]
Percy A. MacMahon, Combinatory analysis. Vol. I, II, Cambridge
University Press, 1915.

[34]
Wojciech Młotkowski, Operator-valued version of conditionally free
product, Studia Math. 153 (2002), no. 1, 13–30.

[35]
James A. Mingo and Roland Speicher, Schwinger-Dyson equations:
classical and quantum, Probab. Math. Statist. 33 (2013), no. 2,

[36]
Sarah Manzel and Michael Schürmann, Non-commutative stochastic
independence and cumulants, Infin. Dimens. Anal. Quantum Probab. Relat. Top.
20 (2017), no. 2, 1750010, 38, arXiv:1601.06779.

[37]
Naofumi Muraki, Monotonic convolution and monotonic
Lévy-Hinčin formula, Preprint, 2000.

[38]
, Monotonic independence, monotonic central limit theorem and
monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat.
Top. 4 (2001), no. 1, 39–58.

[39]
, The five independences as quasi-universal products, Infin.
Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), no. 1, 113–134.

[40]
, The five independences as natural products, Infin. Dimens.
Anal. Quantum Probab. Relat. Top. 6 (2003), no. 3, 337–371.

[41]
Naofumi Muraki, Private communication, 2013.

[42]
Alexandru Nica, A one-parameter family of transforms, linearizing
convolution laws for probability distributions, Comm. Math. Phys.
168 (1995), no. 1, 187–207.

[43]
Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free
probability, London Mathematical Society Lecture Note Series, vol. 335,
Cambridge University Press, Cambridge, 2006.

[44]
Jean-Christophe Novelli and Jean-Yves Thibon, Construction de trigèbres
dendriformes, C. R. Math. Acad. Sci. Paris 342 (2006), no. 6,

[45]
J. K. Percus, Correlation inequalities for Ising spin lattices, Comm.
Math. Phys. 40 (1975), 283–308.

[46]
Mihai Popa, A combinatorial approach to monotonic independence over a
${C}^{*}$-algebra, Pacific J. Math. 237 (2008), 299–325.

[47]
Frédéric Patras and Manfred Schocker, Trees, set compositions and the
twisted descent algebra, J. Algebr. Combin. 28 (2008), 3–23.

[48]
Christophe Reutenauer, Free Lie algebras, London Mathematical Society
Monographs New Series, vol. 7, Clarendon Press, Oxford University Press, New
York, 1993.

[49]
John Riordan, An introduction to combinatorial analysis, Wiley
Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York;
Chapman & Hall, Ltd., London, 1958.

[50]
Gian-Carlo Rota, On the foundations of combinatorial theory I. Theory
of Möbius functions, Z. Wahrscheinlichkeitstheorie 2 (1964),
no. 4, 340–368.

[51]
Gian-Carlo Rota and B. D. Taylor, The classical umbral calculus, SIAM J.
Math. Anal. 25 (1994), no. 2, 694–711.

[52]
M. P. Schützenberger, Contribution aux applications statistiques de
la théorie de l'information, Publ. Inst. Statist. Univ. Paris 3
(1954), no. 1-2, 3–117.

[53]
Barry Simon, The statistical mechanics of lattice gases. Vol. I,
Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993.

[54]
Michael Skeide, Independence and product systems, Recent Developments in
Stochastic Analysis and Related Topics, Fields Inst. Commun., World
Scientific Publication, Hackensack, NJ, 2004, pp. 420–438.

[55]
Louis Solomon, A Mackey formula in the group ring of a Coxeter
group, J. Algebra 41 (1976), no. 2, 255–264.

[56]
Roland Speicher, Multiplicative functions on the lattice of noncrossing
partitions and free convolution, Math. Ann. 298 (1994), no. 4,

[57]
, On universal products, Free probability theory (Waterloo,
ON, 1995), Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI,
1997, pp. 257–266.

[58]
, Combinatorial theory of the free product with amalgamation and
operator-valued free probability theory, Mem. Amer. Math. Soc. 132
(1998), no. 627, x+88.

[59]
Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge
Studies in Advanced Mathematics, vol. 49, Cambridge University Press,
Cambridge, 1997.

[60]
, Enumerative combinatorics. Vol. 2, Cambridge Studies in
Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999.

[61]
Roland Speicher and Wilhelm von Waldenfels, A general central limit
theorem and invariance principle, QP-PQ Quantum Probability and Related
Topics IX, World Scientific, Singapore, 1994, pp. 371–387.

[62]
Roland Speicher and Reza Woroudi, Boolean convolution, Free probability
theory (Waterloo, ON, 1995), Fields Inst. Commun., vol. 12, Amer. Math.
Soc., Providence, RI, 1997, pp. 267–279.

[63]
Robert C. Thompson, Cyclic relations and the Goldberg coefficients in
the Campbell-Baker-Hausdorff formula, Proc. Amer. Math. Soc.
86 (1982), no. 1, 12–14.

[64]
Andy Tonks, Relating the associahedron and the permutohedron, Operads:
Proceedings of Renaissance Conferences (Hartford, CT/Luminy,
1995), Contemp. Math., vol. 202, Amer. Math. Soc., Providence, RI, 1997,
pp. 33–36.

[65]
Alexander Van-Brunt and Matt Visser, Simplifying the Reinsch algorithm
for the Baker-Campbell-Hausdorff series, J. Math. Phys. 57
(2016), no. 2, 023507, 19, arXiv:1501.05034.

[66]
Hans van Leeuwen and Hans Maassen, An obstruction for $q$-deformation
of the convolution product, J. Phys. A 29 (1996), no. 15,

[67]
Dan Voiculescu, Symmetries of some reduced free product
$C^\ast$-algebras, Operator algebras and their connections with topology
and ergodic theory (Buşteni, 1983), Lecture Notes in Math., vol. 1132,
Springer, Berlin, 1985, pp. 556–588.

[68]
, Addition of certain noncommuting random variables, J. Funct.
Anal. 66 (1986), no. 3, 323–346.

[69]
Wilhelm von Waldenfels, An approach to the theory of pressure broadening
of spectral lines, Probability and information theory, II, Springer, Berlin,
1973, pp. 19–69. Lecture Notes in Math., Vol. 296.

[70]
, Interval partitions and pair interactions, Séminaire de
Probabilités, IX, Springer, Berlin, 1975, pp. 565–588. Lecture Notes in
Math., Vol. 465.

[71]
Piotr Zwiernik, $L$-cumulants, $L$-cumulant embeddings and algebraic
statistics, J. Algebr. Stat. 3 (2012), no. 1, 11–43.