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 number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.

The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.

The number of join irreducibles minus the rank of a lattice. A lattice is join-extremal, if this statistic is $0$.