The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.

The number of simple modules with projective dimension at most 1.

The sum of the number of descents and the number of recoils of a permutation. This statistic is the sum of [[St000021]] and [[St000354]].

The number of inversions of distance at most 2 of a permutation. An inversion of a permutation $\pi$ is a pair $i < j$ such that $\sigma(i) > \sigma(j)$. Let $j-i$ be the distance of such an inversion. Then inversions of distance at most 1 are then exactly the descents of $\pi$, see [[St000021]]. This statistic counts the number of inversions of distance at most 2.

The reduced reflection length of the permutation. Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is $$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$ In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].

The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). In symbols, for a permutation $\pi$ this is $$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 1 \leq i_1,\ldots,i_k \leq n\},$$ where $\tau_a = (a,a+1)$ for $1 \leq a \leq n$ and $n+1$ is identified with $1$. Put differently, this is the number of cyclically simple transpositions needed to sort a permutation.

The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. An admissible inversion of a permutation $\sigma$ is a pair $(\sigma_i,\sigma_j)$ such that 1. $i < j$ and $\sigma_i > \sigma_j$ and 2. either $\sigma_j < \sigma_{j+1}$ or there exists a $i < k < j$ with $\sigma_k < \sigma_j$. This version was introduced by John Shareshian and Michelle L. Wachs in [1], for a closely related version, see [[St000463]].

The prefix exchange distance of a permutation. This is the number of star transpositions needed to write a permutation. In symbols, for a permutation $\pi\in\mathfrak S_n$ this is $$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n\},$$ where $\tau_a = (1,a)$ for $2 \leq a \leq n$. [1, Lem. 2.1] shows that the this length is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$. One may find in [2] explicit formulas for its generating function and a combinatorial proof that it is asymptotically normal.