The number of almost positive roots of a finite Cartan type. A root in the root system of a Cartan type is almost positive if it is either positive or simple negative. These are known to be in bijection with cluster variables in the cluster algebra of the given Cartan type, see [1]. This is also equal to the sum of the degrees of the fundamental invariants of the group.

The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J.

The number of posets with the same order polynomial. The order polynomial of a poset $P$ is the polynomial $S$ such that $S(m)$ is the number of order-preserving maps from $P$ to $\{1,\dots,m\}$. See sections 3.12 and 3.15 of [1].

The size of a partition. This statistic is the constant statistic of the level sets.

The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]

The least common multiple of the parts of the partition.

The product of the parts of an integer partition.