The number of elements which do not have a complement in the lattice. A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.

Half the total irregularity of a graph. This is half the sum of the absolute values of the degree differences of all pairs of vertices: $$ \frac{1}{4}\sum_{u,v} |d_u-d_v| $$ It is easy to show by induction on the number of edges that this is an integer.

The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.

The number of edges that can be added without increasing the clique number.

The number of edges that can be added without increasing the chromatic number.

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.

The charge of a standard tableau.

The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. Let $n(G)$ be the number of vertices of a graph $G$, $m(G)$ be its number of edges and let $\alpha(G)$ be its independence number, [[St000093]]. Turán's theorem is that $m(G) \geq m(T^c(n(G), \alpha(G)))$ where $T^c(n, r)$ is the complement of the Turán graph. This statistic records the difference.

The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$.

The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. A 3-Catalan path is a lattice path from $(0,0,0)$ to $(n,n,n)$ consisting of steps $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ such that for each point $(x,y,z)$ on the path we have $x \geq y \geq z$. Its first and last coordinate projections, denoted by $D_{xy}$ and $D_{yz}$, are the Dyck paths obtained by projecting the Catalan path onto the $x,y$-plane and the $y,z$-plane, respectively. For a given Dyck path $D$ this is the number of Catalan paths $C$ such that $D_{xy}(C) = D_{yz}(C) = D$. If $D$ is of semilength $n$, $r_i(D)$ denotes the number of downsteps between the $i$-th and $(i+1)$-st upstep, and $s_i(D)$ denotes the number of upsteps between the $i$-th and $(i+1)$-st downstep, then this number is given by $\prod\limits_{i=1}^{n-1} \binom{r_i(D) + s_i(D)}{r_i(D)}$.