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

The number of degree 2 vertices of a graph. A vertex has degree 2 if and only if it lies on a unique maximal path.

The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$. This statistic counts the $3$-rim hooks that are removed in this process to obtain a $3$-core.

The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].

The exponent of a binary word. This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called '''frequency'''.

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 semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.

The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].

Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$.