The breadth of a permutation. According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$ According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$. By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$. By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this. According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.

The minimum jump of a permutation. This is $\min_i |\pi_{i+1}-\pi_i|$, see [1].

The number of occurrences of the vincular pattern |12-3 in a permutation. This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.

The number of occurrences of the Hertzsprung pattern 132 in a permutation. A Hertzsprung occurrence of the pattern $\tau=(\tau_1,\dots,\tau_k)$ in a permutation $\pi$ is a factor $\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of $\pi$ such that $\pi_{i+j-1} - \tau_j$ is constant for $1\leq j\leq k$.

The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. A graph is chordal if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. A graph is an interval graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ is an edge and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

The minimal number of edges to add or remove to make a graph edge transitive. A graph is edge transitive, if for any two edges, there is an automorphism that maps one edge to the other.

The minimal number of edges to add or remove to make a graph a cograph. A cograph is a graph that can be obtained from the one vertex graph by complementation and disjoint union.