The minimum rank of a graph. The minimum rank of a simple graph G is the smallest possible rank over all symmetric real matrices whose entry in row $i$ and column $j$ (for $i\neq j$) is nonzero whenever $\{i, j\}$ is an edge in $G$, and zero otherwise.

Difference between largest and smallest parts in a partition.

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

The sum of the dimension of Ext^i(D(A),A) for i=1,...,g when g denotes the global dimension of the corresponding LNakayama algebra.

The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. Also the number of simple modules that are isolated vertices in the sense of 4.5. (4) in the reference.

The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module. For the first 196 values, the statistic also gives the number of indecomposable non-projective modules $X$ such that $\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.