The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.

The cyclic holeyness of a permutation. For $S\subset [n]:=\{1,2,\dots,n\}$ let $\delta(S)$ be the number of elements $m\in S$ such that $(m\bmod n)+1\notin S$. For a permutation $\pi$ of $[n]$ the cyclic holeyness of $\pi$ is $$\max_{S\subset [n]} (\delta(\pi(S))-\delta(S)).$$

The number of 1-protected nodes of a rooted tree. This is the number of nodes with minimal distance one to a leaf.

The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].

The number of left outer peaks of a permutation. A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$. In other words, it is a peak in the word $[0,w_1,..., w_n]$. This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

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 cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$. Clearly, every cycle of $\pi$ contains as many peaks as valleys.

The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.

The number of descents of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. The word $w$ has a descent at position $i$ if $w_i > w_{i+1}$.