The difference of the maximal and the minimal degree in a graph. The graph is regular if and only if this statistic is zero.

The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.

The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$. Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].

The minimal period of a binary word. This is the smallest natural number $p$ such that $w_i=w_{i+p}$ for all $i\in\{1,\dots,|w|-p\}$.

The maximal difference between two consecutive entries of a permutation. This is given, for a permutation $\pi$ of length $n$, by $$\max\{ | \pi(i) - \pi(i+1) | : 1 \leq i < n \}.$$

The cardinality of the support of a permutation. A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$. The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$. Thus, the connectivity set is the complement of the support.

The depth of a permutation. This is given by $$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$. Permutations with depth at most $1$ are called ''almost-increasing'' in [5].

The sum of the descent differences of a permutations. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$ See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.

The degree of the graph. This is the maximal vertex degree of a graph.

Maximum difference of elements in cycles. Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$. The statistic is then the maximum of this value over all cycles in the permutation.