The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].

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 maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].

The bandwidth of a graph. The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$. We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.

The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].