The number of connected elements in the Coxeter group corresponding to a finite Cartan type. Let $(W, S)$ be a Coxeter system. Then, according to [1], the connectivity set of $w\in W$ is the cardinality of $S \setminus S(w)$, where $S(w)$ is the set of generators appearing in any reduced word for $w$. For $A_n$, this is [2], for $B_n$ this is [3] and for $D_n$ this is [4].

The total irregularity of a graph. This is the sum of the absolute values of the degree differences of all pairs of vertices: $$ \frac{1}{2}\sum_{u,v} |d_u-d_v| $$

The number of pairs of vertices of different degree in a graph.

The leading coefficient of the rook polynomial of an integer partition. Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.

The number of ways to place as many non-attacking rooks as possible on a Ferrers board.

The number of partitions interlacing the given partition.

Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial $$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$ The statistic of the degree of this polynomial. For example, the partition $(3, 2, 1, 1, 1)$ gives $$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$ which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$. This is the same as the number of unordered pairs of different parts, which follows from: $$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$

The Ore degree of a graph. This is the maximal Ore degree of an edge, which is the sum of the degrees of its two endpoints.

The Albertson index of a graph. This is $\sum_{\{u,v\}\in E} |d(u)-d(v)|$, where $E$ is the set of edges and $d_v$ is the degree of vertex $v$, see [1]. In particular, this statistic vanishes on graphs whose components are all regular, see [2].

The Padmakar-Ivan index of a graph. For an edge $e=(u, v)$, let $n_{e, u}$ be the number of edges in a graph $G$ induced by the set of vertices $\{w: d(u, w) < d(v, w)\}$, where $d(u,v)$ denotes the distance between $u$ and $v$. Then the PI-index of $G$ is $$\sum_{e=(u,v)} n_{e, u} + n_{e, v}.$$