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 area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$. 2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.

Gives the vector space dimension of the homomorphism space between J^2 and J^2.

The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.

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 edges minus the number of vertices plus 2 of a graph. When G is connected and planar, this is also the number of its faces. When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.

The sum of the dimension of Ext^i(D(A),A) for i=1,...,g when g denotes the global dimension of the corresponding LNakayama algebra.

The diagonal index (content) of a partition. The '''diagonal index''' of the cell at row $r$ and column $c$ of a partition is $c - r$; this is sometimes called the '''content''' of the cell. The '''diagonal index of a partition''' is the sum of the diagonal index of each cell of the partition.

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 $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I. See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.