The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details. The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).

The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.

The number of induced paths on four vertices in a graph.

Half of the Albertson index of a graph. This is $\frac{1}{2}\sum_{\{u,v\}\in E}|d(u)-d(v)|$, where $E$ is the set of edges and $d_v$ is the degree of vertex $v$, see [1]. In particular, this statistic vanishes on graphs whose components are all regular, see [2].

The number of prime nodes in the modular decomposition of a graph.

The differential of a graph. The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$. The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$. The differential of a graph is the maximal differential of a set of vertices.