The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1]. This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for increasing trees, whose generating function is $f(x) = -\log(1-x)$, see [1, sec. 9.1] Fix a coloring of $\{1,2, \ldots, n\}$ so that $\lambda_i$ are colored with the $i$th color. This statistic gives the number of increasing trees on this colored set of vertices so that no leaf has the same color as its parent. (An increasing tree is a rooted tree on the vertex set $\{1,2, \ldots, n\}$ with the property that any child of $i$ is greater than $i$.)

The chromatic discriminant of a graph. The chromatic discriminant $\alpha(G)$ is the coefficient of the linear term of the chromatic polynomial $\chi(G,q)$. According to [1], it equals the cardinality of any of the following sets: (1) Acyclic orientations of G with unique sink at $q$, (2) Maximum $G$-parking functions relative to $q$, (3) Minimal $q$-critical states, (4) Spanning trees of G without broken circuits, (5) Conjugacy classes of Coxeter elements in the Coxeter group associated to $G$, (6) Multilinear Lyndon heaps on $G$. In addition, $\alpha(G)$ is also equal to the the dimension of the root space corresponding to the sum of all simple roots in the Kac-Moody Lie algebra associated to the graph.