The number of non-empty open intervals in a poset. An open interval $(x, y)$, for $x < y$, is the set of elements $\{z | x < z < y\}$.

The difference in the number of possibilities of choosing a pair of negative eigenvalues and the signature of a graph. Let $n^+(G)$, $n^0(G)$ and $n^-(G)$ denote the number of positive, zero, and negative eigenvalues of $G$. The signature of a graph $s(G)$ is $n^+(G) - n^-(G)$. This statistic records $\binom{n^-(G)+1}{2} - n^+(G) = \binom{n^-(G)}{2} - s(G)$. In [1], it is conjectured to be non-negative.

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$

The major index east count of a Dyck path. The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]]. The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.

The number of doubly irreducible elements of a lattice. An element $d$ of a lattice $L$ is '''doubly irreducible''' if it is both join and meet irreducible. That means, $d$ is neither the least nor the greatest element of $L$ and if $d=x\vee y$ or $d=x\wedge y$, then $d\in\{x,y\}$ for all $x,y\in L$. In a finite lattice, the doubly irreducible elements are those which cover and are covered by a unique element.

The number of prime labellings of a graph. A prime labelling of a graph is a bijective labelling of the vertices with the numbers $\{1,\dots, |V(G)|\}$ such that adjacent vertices have coprime labels.