The number of orbits of reflections of a finite Cartan type. Let $W$ be the Weyl group of a Cartan type. The reflections in $W$ are closed under conjugation, and this statistic counts the number of conjugacy classes of $W$ that are reflections. It is well-known that there are either one or two such conjugacy classes.

The number of full-support reflections in the Weyl group of a finite Cartan type. A reflection has full support if any (or all) reduced words for it in simple reflections use all simple reflections. This number is given by $\frac{nh}{|W|}d_1^*\cdots d_{n-1}^*$ where $n$ is the rank, $h$ is the Coxeter number, $W$ is the Weyl group, and $d_1^* \geq \ldots \geq d_{n-1}^* \geq d_n^* = 0$ are the codegrees of the Weyl group of a Cartan type.

The size of the mutation class of quivers of given type.

The largest coefficient in the highest root in the root system of a Cartan type.

The minimal size of a base for the Weyl group of the Cartan type. A base of a permutation group is a set $B$ such that the pointwise stabilizer of $B$ is trivial. For example, a base of the symmetric group on $n$ letters must contain all but one letter. Any base has at least $\log |G|/n$ elements, where $n$ is the degree of the group, i.e., the size of its domain.

The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.

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.

The number of maximal antichains of minimal length in a poset.

The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$

The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1].