The determinant of the Cartan matrix. This is also the order of the center of the corresponding simply connected group.

The number of minuscule dominant weights in the weight lattice of a finite Cartan type. In short, this is the number of simple roots that appear with multiplicity one in the hightest root of the root system. By definition, a weight $\lambda \neq 0$ in the weight lattice is '''dominant''' if $\langle \lambda, \alpha\rangle \geq 0$ for all simple roots $\alpha$ and a dominant weight is '''minuscule''' if $\langle \lambda, \beta\rangle \in \{0,\pm 1\}$ for all roots $\beta$. Since $\langle \lambda, \alpha\rangle \in \{0,1\}$ for simple roots $\alpha$, we have that $\lambda$ is minuscule if and only if it is fundamental and $\langle \lambda, \rho\rangle = 1$ for the unique highest root $\rho$. The number of minuscule dominant weights is one less than the determinant of the Cartan matrix [[St000821]]. They index the nontrivial minuscule representations, see [1].

The minimal height of a column in the parallelogram polyomino associated with the Dyck path.

The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.

The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. The deck of a graph is the multiset of induced subgraphs obtained by deleting a single vertex. The graph reconstruction conjecture states that the deck of a graph with at least three vertices determines the graph. This statistic is only defined for graphs with at least two vertices, because there is only a single graph of the given size otherwise.

The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.

The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$. This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid. An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].

The number of factors of a lattice as a Cartesian product of lattices. Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.

The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.