The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.

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 $w_i > w_{i+1}$.

The number of outer peaks of a permutation. An 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 $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$. In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.

The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].

The number of inner corners of a skew partition.

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

The number of factors DDU in a Dyck path.

The number of big descents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$. The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.