The number of lucky cars of the parking function. A lucky car is a car that was able to park in its prefered spot. The generating function, $$ q\prod_{i=1}^{n-1} (i + (n-i+1)q) $$ was established in [1].

The number of characters of the symmetric group whose value on the partition is positive.

The multiplicity of the largest part of an integer partition.

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 largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.