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 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.

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.

Number of indecomposable reflexive modules in the corresponding Nakayama algebra.

The position of the last up step in a Dyck path.

The major index north 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 north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.

The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that $$ \alpha\pi\alpha^{-1} = \pi^2.$$