May 31, 2015. Cumulants are like the moments of a probability distribution, but better. I discuss a nice technique for calculating them in terms of Möbius functions on partition lattices, and use it to prove a famous result about Gaussian moments attributed to Wick/Isserlis. These are more or less just my notes on a paper by Terry Speed.

Introduction

Prerequisites: basic probability theory, some mathematical maturity.

For a continuous random variable $X$, the moment-generating function $M_X(t)$ is defined by

\[M_X(t) = \mathbb{E}(e^{tX}) = \sum_{k=0}^\infty \mathbb{E}(X^k)\frac{t^k}{k!} = \sum_{k=0}^\infty m_k\frac{t^k}{k!}.\]

The Taylor coefficient $m_k$ is called the $k$th moment of the distribution. Integral combinations of moments give us the mean, variance, skew, kurtosis, etc, of the distribution. A closely related object is the cumulant-generating function $g_X(t)$, given by

\[g_X(t) = \log M_X(t) = \sum_{k=0}^\infty \kappa_k\frac{t^k}{k!}.\]

The Taylor coefficient $\kappa_k$ is called the $k$th cumulant of $X$. Cumulants are used in statistics, but as we will see, also have interesting applications to diagrammatic methods in physics. This first post is going to explain a nice formalism for calculating joint cumulants using set partitions due to Terry Speed (1983), and follows Speed’s paper closely (though I include some details Speed omits). In a sequel, I hope to explain how cumulants are used in statistical physics and quantum field theory.

Cumulants

Joint cumulants are cumulants for jointly distributed random variables. Say we have $m$ random variables $X_1, \ldots, X_m$. Use the vector notation $\mathbf{r} = (r_1, \ldots, r_m) \in \mathbb{Z}^m$, with $\mathbf{r} \geq 0$ meaning each $r_i \geq 0$, and

\[\begin{align*} \mathbf{X}^\mathbf{r} & = X_1^{r_1}\cdots X_m^{r_m}, \quad \mathbf{\theta}^\mathbf{r} = \theta_1^{r_1}\cdots \theta_m^{r_m}, \\ \mathbf{r}! & = r_1!\cdots r_m!, \quad r = |\mathbf{r}|_1 = r_1 + \cdots+ r_m. \end{align*}\]

By analogy with the single-variable case, we can define the joint cumulants via

\[\sum_{\mathbf{r} \geq 0} \kappa_\mathbf{r}\frac{\mathbf{\theta}^\mathbf{r}}{\mathbf{r}!} = \log \sum_{\mathbf{r} \geq 0} \mathbb{E}(X^\mathbf{r})\frac{\mathbf{\theta}^\mathbf{r}}{\mathbf{r}!}.\]

Quite simply, joint cumulants are the coefficients of the multivariable Taylor expansion of log of the multivariable moment generating function. It’s convenient to have a way of writing the cumulant $\kappa_\mathbf{r}$ with the random variables it depends on made explicit:

\[\kappa_\mathbf{r} = \mathcal{C}(\overset{r_1 \text{ times}}{\overbrace{X_1, \ldots, X_1}}, \ldots, \overset{r_m \text{ times}}{\overbrace{X_m, \ldots, X_m}}) = \mathcal{C}(X'_1, \ldots, X'_{r}).\]

Here, the $X_i$ are the specific jointly distributed random variables we started with, while the $X’_i$ are possibly repeated RV inputs to $\mathcal{C}$. Setting the $X’_i$ as above, it turns out that we can define

\[\begin{align*} \kappa_\mathbf{r} &= \mathcal{C}(X'_1, \ldots, X'_{r}) \\ &= \sum_{\sigma\in P_r}(-1)^{b(\sigma) - 1}(b(\sigma)-1)!\prod_{a=1}^{b(\sigma)} \mathbb{E}\left(\prod_{i\in\sigma_a} X'_i\right), \end{align*}\]

where $P_{r}$ is the set of partitions of the set $[r] := {1, \ldots, r}$, and $\sigma \in P_r$ splits into $b := b(\sigma)$ disjoint blocks $\sigma_1, \ldots, \sigma_b$ satisfying

\[\bigcup_i \sigma_i = \{1, \ldots, r\}, \quad \sigma_i \cap \sigma_j = \varnothing \text{ for } i\neq j.\]

It’s not at all obvious that this definition of joint cumulant is useful, or even equivalent, but we’ll show that both are true below.

Before we go on, we do a calculation to illustrate the utility of the second definition. For three jointly distributed random variables, $\kappa_{111} = \mathcal{C}(X_1, X_2, X_3)$ can be calculated in terms of expectations using the alternative definition. We can simplify our partitions by concatenating numbers in the same block and separating blocks with bars:

\[P_3 = \{123, 1|23, 2|13, 3|12, 1|2|3\}.\]

The corresponding blocks numbers are ${1, 2, 2, 2, 3}$. We sum over the partitions with the prefactor $(-1)^{b(\sigma)-1}(b(\sigma)-1)!$ and take the product of expectations over products within blocks:

\[\begin{align*} \mathcal{C}(X_1, X_2, X_3) & = \mathbb{E}(X_1X_2X_3) - \mathbb{E}(X_1)\mathbb{E}(X_2X_3) \\ & \qquad - \mathbb{E}(X_2)\mathbb{E}(X_1X_3) - \mathbb{E}(X_3)\mathbb{E}(X_1X_2) + 2\mathbb{E}(X_1)\mathbb{E}(X_2)\mathbb{E}(X_3). \end{align*}\]

We can set $X_1 = X_2 = X_3 = X$ to obtain a result for the single-variable cumulant $\kappa_3$:

\[\kappa_3 = \mathbb{E}(X^3) - 3\mathbb{E}(X)\mathbb{E}(X^2) + 2\mathbb{E}(X)^3 = m_3 - 3m_1m_2 + 2m_1^3.\]

Partition lattices

Let’s be more explicit about partitions. A partition $\sigma$ of a set $S$ is a collection of disjoint subsets of $S$ whose union is $S$, i.e.

\[\sigma = \{\sigma_i\}_{i\in I}, \quad \bigcup_{i\in I} \sigma_i = S, \quad \sigma_i \cap \sigma_j = \varnothing \text{ for } i\neq j.\]

We call the $\sigma_i$ the blocks of $\sigma$. We label the collection of all such partitions $P(S)$. If $\sigma, \tau \in P(S)$, and each $\sigma_i \subseteq \tau_j$ for some $j$ (so each block of $\sigma$ is in a block of $\tau$) we say that $\sigma$ is finer than $\tau$, or $\tau$ is coarser than $\sigma$, and write $\sigma \leq \tau$. This relation turns $P(S)$ into a special algebraic object called a lattice.

A lattice $(\mathcal{L}, \leq)$ is a partially ordered set where any two elements have a greatest lower bound and a least upper bound, also called the meet and join respectively. We recall that a partial order $\leq$ is reflexive, transitive and antisymmetric:

\[\begin{align*} (\textrm{R}) \,\,&x \leq x, \\ (\textrm{T}) \,\,&x \leq y, \quad y\leq z \quad\Longrightarrow\quad x \leq z, \\ (\textrm{A}) \,\,&x \leq y, \quad y\leq x \quad\Longrightarrow\quad x = y. \end{align*}\]

The meet $x \wedge y$ and join $x \vee y$ of $x, y \in \mathcal{L}$ are defined (if they exist) as

\[\begin{align*} x \vee y & := \max \{z : z \leq x, z \leq y\} \\ x \wedge y & := \min \{z : x \leq z, y \leq z\}. \end{align*}\]

In a lattice, the meet and join exist for arbitrary elements. A familiar example of a lattice is the two-element boolean algebra ${0, 1}$ with the usual partial order $0 \leq 1$, and meet and join are the logical operations “and” and “or” (hence the notation). Another example is the power set $2^A$ of any set $A$ ordered by the subset relation, where meet and join are intersection and union.

So, I claim that using the “finer than” relation $\leq$ on $P(S)$ turns it into a lattice. First of all, it’s clear that $\leq$ is a partial order since it obviously satisfies (R), (T) and (A). The meet of $\sigma$ and $\tau$ is the coarsest partition of $S$ finer than both, which is easily seen to be the non-empty intersections of the blocks of $\tau$ and $\sigma$. For instance, in $P_3$,

\[1|23 \wedge 12|3 = 1|2|3.\]

Thought of a different way, it is the “union” of the bars. Similarly, the join is the “intersection” of bars, so

\[1|23 \vee 12|3 = 123.\]

We can picture any finite partial order with a graph called a Hasse diagram. These are graphs with no horizontal edges, and an edge $xy$ ($x$ higher) indicates that $y < x$ and there are no $z$ such that $y < z < x$. Each Hasse diagram gives a partial order on the vertices, but not necessarily a lattice. For instance, in a disconnected diagram, vertices in different components cannot have a meet or join.

Exercise 1. (a) Draw the partition lattice for $P_4$. Solution. (b) Find a partial order with a connected Hasse diagram which is not a lattice. (c) Find the smallest answer to (b), and prove that this is so.

Zeta and Möbius functions

Given a finite partially ordered set $(A, \leq)$, there are two associated functions that will prove useful. The first is the enigmatically named zeta function $\zeta_A: A\times A \to A$, defined by

\[\zeta_A(x, y) = \begin{cases} 1 & x \leq y \\ 0 & \text{else.} \end{cases}\]

The second is the Möbius function $\mu_A$, which can be defined as

\[\mu_A(x, y) = \begin{cases} 1 & x = y \\ -\sum_{x\leq z < y} \mu_A(x, z) & x &< y \\ 0 & \text{else.} \end{cases}\]

We can think of the Möbius function as returning $1$ for equal arguments, $0$ for arguments which are not comparable according to $\leq$, and otherwise walking down the path in the Hasse diagram from $y$ to $x$, adding $-\mu_A(x, z)$ for each vertex $z$ it visits along the way (including $x$ but not $y$). In fact, thinking of $\mu_A$ and $\zeta_A$ as matrices

\[\mu_{xy} = \mu_A(x, y), \quad \zeta_{xy} = \zeta_A(x, y),\]

we have $\mu\zeta = \zeta\mu = I$. Let’s check. First of all,

\[\begin{align*} (\mu\zeta)_{xz} & = \sum_y \mu_A(x, y)\zeta_A(y, z) = \sum_{y\leq z} \mu_A(x, y) = \sum_{x\leq y< z}\mu_A(x, y) + \mu_A(x, z). \end{align*}\]

If $x = z$, then the summation vanishes (no $y$ terms) and we are left with $\mu_A(x, z) = 1$. If $x > z$ or they are not comparable, then both terms vanish. Finally, if $x < z$, then

\[(\mu\zeta)_{xz} = \sum_{x\leq y< z}\mu_A(x, y) + \mu_A(x, z) = -\mu_A(x, z) + \mu_A(x, z) = 0.\]

It follows that $\mu\zeta = I$. By linear algebra, $\zeta\mu = I$. Now suppose we have a function $f: A \to \mathbb{R}$, and define the “partial sum” by

\[F(x) = \sum_{y\leq x}f(y) = \sum_{y}f(y)\zeta_A(y, x) = \sum_{y}f_y\zeta_{y, x},\]

thinking of $f_x = f(x)$ as a row matrix. Thinking of $F_x = F(x)$ the same way, we have $F = f\zeta$. Hence,

\[f = f I = f \zeta\mu = F\mu.\]

This is Möbius inversion: using the Möbius function to go back and forth between a function on the lattice and its partial sum. It turns out that $\mu_A$ is often easy to calculate. In the case of lattice partitions, it turns out that

\[\mu_{P(S)}(\sigma, \hat{S}) = (-1)^{b(\sigma)-1}(b(\sigma)-1)!\]

where $\hat{S} = {S}$, the partition of $S$ into a single block.

Exercise 2. (a) Show that for a finite poset and any $x, y$,

\[0 = \sum \{\mu(w,z) : x \leq w \leq z \leq y \}.\]

Conclude that

\[\mu_{A}(x, y) = -\sum_{x < z \leq b} \mu_A(z, b).\]

(b) Using part (a) and induction on $b(\sigma)$, prove that $\mu_{P(S)}(\sigma,\hat{S}) = (-1)^{b(\sigma)-1}(b(\sigma)-1)!$ for any finite set $S$.

Cumulants via Möbius inversion

We now return to our lattice of interest, the partition lattice $P_r$. We would like to use Möbius inversion to show that the definition of cumulants based on partitions, $\mathcal{C}$, is equivalent to the standard definition in terms of the power series, $\kappa$. Consider a partition $\sigma \in P_r$, written in terms of its $b$ blocks $\sigma = \sigma_1 | \cdots | \sigma_b$. Write

\[\bar{\kappa}_\sigma = \prod_{a=1}^b \kappa_{\mathbf{r}(\sigma_a)},\]

where $\mathbf{r}(\sigma_a) = (r_i(\sigma_a))$ is an $r$-vector defined by $r_i(\sigma_a) = 1$ if $i \in \sigma_a$ and $0$ otherwise. If $\sigma = 12|3|4$ for instance, then

\[\bar{\kappa}_\sigma = \kappa_{1100}\kappa_{0010}\kappa_{0001}.\]

If we exponentiate the multivariable Taylor expansion defining $\kappa$, we get

\[\begin{align*} \sum_{\mathbf{r} \geq 0} \mathbb{E}(X^\mathbf{r})\frac{\mathbf{\theta}^\mathbf{r}}{\mathbf{r}!} & = \exp\left[ \sum_{\mathbf{r} \geq 0} \kappa_\mathbf{r}\frac{\mathbf{\theta}^\mathbf{r}}{\mathbf{r}!}\right]\\&=\sum_{n\geq 0}\frac{1}{n!}\left[ \sum_{\mathbf{r} \geq 0} \kappa_\mathbf{r}\frac{\mathbf{\theta}^\mathbf{r}}{\mathbf{r}!}\right]^n\\ & = \sum_{n\geq 0}\frac{1}{n!}\sum_{\mathbf{r}_1,\ldots, \mathbf{r}_n \geq 0} \kappa_{\mathbf{r}_1}\cdots \kappa_{\mathbf{r}_n}\frac{\mathbf{\theta}^{\mathbf{r}_1 + \cdots +\mathbf{r}_n}}{\mathbf{r}_1!\cdots \mathbf{r}_n!}. \end{align*}\]

Equating coefficients of $\mathbf{\theta}^\mathbf{r}$, we see that

\[\mathbb{E}(X^\mathbf{r}) = \sum_{n\geq 0}\sum_{\mathbf{r}_1+\cdots+\mathbf{r}_n = \mathbf{r}} \frac{1}{n!}\prod_{k=1}^n \frac{\kappa_{\mathbf{r}_k}}{\mathbf{r}_k!}.\]

The summation is effectively over integer partitions of $\mathbf{r}$, with $n$ the number of blocks. These are genuine integer partitions, not just partitions of the index set. However, matters are simplified if we specialise to the case where $r_i \leq 1$, i.e., at most linear powers of a random variable appear in the expectation $\mathbb{E}(X^\mathbf{r})$. Then $\mathbf{r}_k! = 1$ and the sum

\[\mathbf{r}_1 + \cdots +\mathbf{r}_n = \mathbf{r}\]

can be thought of as a partition of the occurrences of $1$ in $\mathbf{r}$ into $n$ blocks, i.e. an element $\sigma \in P_r$ with $b(\sigma) = n$. Reordering gives a factor of $n!$, but this is nicely cancelled by the $1/n!$ in the summation over the number of blocks, so we end up with

\[\begin{align*} \mathbb{E}(X^\mathbf{r}) & = \sum_{n\geq 0}\sum_{\mathbf{r}_1+\cdots+\mathbf{r}_n = \mathbf{r}} \frac{1}{n!}\prod_{k=1}^n \kappa_{\mathbf{r}_k} = \sum_{\sigma \in P_r}\prod_{k=1}^n \kappa_{\mathbf{r}(\sigma_k)} = \sum_{\sigma \in P_r}\bar{\kappa}_{\sigma}. \end{align*}\]

Now consider an arbitrary partition of the original $m$ random variables into $b$ blocks, $\tau \in P_m$ with $b(\tau) = b$. Applying the foregoing result, we get

\[\begin{align*} \prod_{a=1}^b \mathbb{E}\left(\prod_{k\in\tau_a} X_k\right) & = \prod_{a=1}^b \sum_{\sigma_a\in P(\tau_a)} \bar{\kappa}_{\sigma_a}. \end{align*}\]

Note that we sum over partitions of the blocks of $\tau$, with $\sigma_a \in P(\tau_a)$ standing for some partition of the set of elements in $\tau_a$. We can assemble the smaller blocks $\sigma_a$ into a partition $\sigma$ of $m$ which is finer than $\tau$, so $\sigma \leq \tau$. It is not hard to see that every finer partition than $\tau$ arises in this way. In other words,

\[\prod_{a=1}^b \mathbb{E}\left(\prod_{k\in\tau_a} X_k\right) = \sum_{\sigma\leq \tau}\bar{\kappa}_{\sigma}.\]

Here’s an example:

\[\mathbb{E}(X_1X_2)\mathbb{E}(X_3X_4) = (\bar{\kappa}_{12} + \bar{\kappa}_{1|2})(\bar{\kappa}_{34} + \bar{\kappa}_{3|4}) = \bar{\kappa}_{12|34} + \bar{\kappa}_{12|3|4} + \bar{\kappa}_{1|2|34} + \bar{\kappa}_{1|2|34}.\]

Now we simply use Möbius inversion on our formula to get an expression for $\bar{\kappa}$:

\[\begin{align*} \bar{\kappa}_{\tau} = \sum_{\sigma \in P_m} \mu_{P_m}(\sigma, \tau)\prod_{a=1}^b \mathbb{E}\left(\prod_{k\in\tau_a} X_k\right). \end{align*}\]

Finally, using the formula for $\mu_{P(S)}(\sigma, \hat{S})$ above, we have

\[\begin{align*} \bar{\kappa}_{\hat{S}} = \kappa_{1\cdots 1} = \sum_{\sigma \in P_m} (-1)^{b(\sigma)-1}(b(\sigma)-1)!\prod_{a=1}^b \mathbb{E}\left(\prod_{k\in\tau_a} X_k\right) \end{align*}\]

as claimed. The general result follows by thinking of any repeated random variables as new (but identically distributed) random variables. For simplicity, we will continue assuming that random variables are distinct.

Isserlis’ theorem

We can apply these results to quickly prove Isserlis’ theorem. Suppose we have a zero mean, multivariate normal $\mathbf{X} = (X_1, \ldots, X_{m}) \sim \mathcal{N}_m(0, \mathbf{\Sigma})$, where $\mathbf{\Sigma}$ is the covariance matrix

\[\mathbf{\Sigma} = [\mathrm{Cov}(X_i, X_j)] = [\mathbb{E}(X_iX_j)].\]

It turns out (see a proof here) that the moment generating function is

\[\begin{align*} M_{\mathbf{X}}(\mathbf{t}) & = \mathbb{E}(e^{\mathbf{t}^{\mathrm{T}}\mathbf{X}}) = \exp\left(\frac{1}{2}\mathbf{t}^{\mathrm{T}}\mathbf{\Sigma}\mathbf{t}\right). \end{align*}\]

So the cumulant generating function is

\[\log M_{\mathbf{X}}(\mathbf{t}) = \frac{1}{2}\mathbf{t}^{\mathrm{T}}\mathbf{\Sigma}\mathbf{t} = \frac{1}{2}\sum_{i,j}\mathbb{E}(X_iX_j)t_i t_j.\]

Comparing to the vector notation, it’s not hard to see that

\[\mathbb{E}(X_iX_j) = \kappa_{\mathbf{r}}\]

where $r_i = r_j = 1$, $r_k = 0$ for $k \notin {i, j}$. Thus, only cumulants of order $2$ are non-vanishing. Using our result for expectations in terms of $\bar{\kappa}$,

\[\begin{align*} \mathbb{E}(X_1\cdots X_m) & = \sum_{\sigma \in P_m} \bar{\kappa}_\sigma = \sum_{\sigma \in W} \bar{\kappa}_\sigma \end{align*}\]

where $W$ is the subset of $P_m$ where each block is a pair, since the product of cumulants associated with any other permutation vanishes. Thus, if $m$ is odd,

\[\mathbb{E}\left(\prod_{i=1}^m X_i\right) = 0.\]

On the other hand, if $m = 2n$, then

\[\mathbb{E}\left(\prod_{i=1}^m X_i\right) = \sum_{\sigma \in W}\prod_{a=1}^n \mathbb{E}(X_{i_a}X_{j_a}), \quad \sigma_a = \{i_a, j_a\}.\]

Leon Isserlis proved these last two results in 1916, using induction. They were proved in a slightly different form by the physicist Gian-Carlo Wick in 1950, so in quantum field theory the same result is called Wick’s theorem. I hope to discuss the theorem of Isserlis/Wick in greater depth in a subsequent post.

A theorem of Leonov and Shiryaev

We now apply our lattice theoretic expression to prove a result due to Leonov and Shiryaev (1959). Consider an array $[X_{ij}]$ of real random variables, $ i \in [m], j \in [n_i]$, so the length of a row may vary. Any partition $\pi \in P_m$ of the row indices partitions the full set $S = {(i, j): i\in[m], j\in[n_i]}$, partitioning the corresponding rows. Denote the induced partition of $S$ by $\tilde{\pi}$. A partition $\sigma \in P(S)$ is decomposable relative to $\pi \in P_m$ if $\sigma \leq \tilde{\pi}$, and indecomposable if $\sigma$ is not decomposable with respect to any row partition other than $\hat{[m]}$. In other words, an indecomposable partition has a block with an element from each row.

The general result is

\[\mathcal{C}\left(\prod_{j_1\in[n_1]}X_{ij_1}, \ldots, \prod_{j_m\in[n_m]}X_{ij_m}\right) = \sum_{\sigma}^* \prod_{a=1}^{b(\sigma)}\mathcal{C}(X_{ij}: (i, j) \in \sigma_a).\]

The asterisk over the sum on the right indicates the restriction to indecomposable partitions of $S$. Let’s prove this. Let $\pi \in P_m$. By our earlier result for cumulants,

\[\prod_{a=1}^{b(\pi)}\mathbb{E}\left(\prod_{i\in\pi_a}\prod_{j_i\in [n_i]} X_{ij_i}\right) = \sum_{\sigma \leq \tilde{\pi}} \prod_{a=1}^{b(\sigma)}\mathcal{C}(X_{ij}: (i,j) \in \sigma_a),\]

where we made the replacements

\[\begin{align*} \mathbb{E}\left(\prod_{k\in\tau_a} X_k\right) \quad & \longrightarrow \quad \mathbb{E}\left(\prod_{i\in\pi_a}\prod_{j_i\in [n_i]} X_{ij_i}\right),\\ \sum_{\sigma\leq \tau}\bar{\kappa}_{\sigma} \quad & \longrightarrow \quad \sum_{\sigma \leq \tilde{\pi}} \prod_{a=1}^{b(\sigma)}\mathcal{C}(X_{ij}: (i,j) \in \sigma_a). \end{align*}\]

The first replacement is just an explicit product over the partition $\tilde{\pi}$, and the second uses the definition of $\bar{\kappa}$, summing over the partitions $\sigma \in P(S)$ decomposable relative to $\pi$. Define $F(\pi)$ and $f(\rho)$ via

\[\begin{align*} F(\pi) & = \sum_{\sigma \leq \tilde{\pi}} \prod_{a=1}^{b(\sigma)}\mathcal{C}(X_{ij}: (i,j) \in \sigma_a) \\ & = \sum_{\rho}\zeta_{P_m}(\rho, \pi)\sum_{\sigma \prec \rho} \prod_{a=1}^{b(\sigma)}\mathcal{C}(X_{ij}: (i,j) \in \sigma_a)\\ & = \sum_{\rho}\zeta_{P_m}(\rho, \pi)f(\rho), \end{align*}\]

where $\sigma \prec \rho$ indicates that $\sigma$ is decomposable with respect to $\rho$ but no finer row partition. Using Möbius inversion over $P_m$, we find

\[f(\pi) = \sum_\rho \mu_{P_m}(\rho, \pi)F(\rho).\]

Set $\pi = \hat{[m]}$. Using this expression, our formula from Exercise 2, and the definition of $\mathcal{C}$, we obtain

\[\begin{align*} \mathcal{C}\left(\prod_{j_1\in[n]_1}X_{ij_1}, \ldots, \prod_{j_m\in[n]_m}X_{ij_m}\right) & = \sum_{\rho}(-1)^{b(\rho) - 1}(b(\rho)-1)!\prod_{a=1}^{b(\rho)} \mathbb{E}\left(\prod_{i\in\rho_a}\prod_{j_i\in [n_i]} X_{ij_i}\right)\\ &= \sum_{\rho}\mu_{P_m}(\rho, \hat{[m]})F(\rho)\\ & = f(\hat{[m]}) \\ & = \sum_{\sigma \prec \hat{[m]}} \prod_{a=1}^{b(\sigma)}\mathcal{C}(X_{ij}: (i,j) \in \sigma_a). \end{align*}\]

Since $\sigma \prec \hat{[m]}$ means that $\sigma$ is decomposable with respect to $\hat{[m]}$ but no finer row partition, by definition it is indecomposable. This completes the proof.

References

• Cumulants and partition lattices (1983), Terry Speed.