The number of occurrences of the vincular pattern |1-23 in a permutation. This is the number of occurrences of the pattern $123$, where the first two matched entries are the first two entries of the permutation. In other words, this statistic is zero, if the first entry of the permutation is larger than the second, and it is the number of entries larger than the second entry otherwise.

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

The greater neighbor of the maximum. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the smallest path leaf label of the binary tree associated to a permutation ([[St000724]]), see also [3].

The number of maximal entries in the last diagonal of the monotone triangle. Consider the alternating sign matrix $$ \left(\begin{array}{rrrrr} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \end{array}\right). $$ The corresponding monotone triangle is $$ \begin{array}{ccccccccc} 5 & & 4 & & 3 & & 2 & & 1 \\ & 5 & & 4 & & 3 & & 1 & \\ & & 5 & & 3 & & 1 & & \\ & & & 5 & & 3 & & & \\ & & & & 4 & & & & \end{array} $$ The first entry $1$ in the last diagonal is maximal, because rows are strictly decreasing and its left neighbour is $2$. Also, the entry $3$ in the last diagonal is maximal, because diagonals from north-west to south-east are weakly decreasing, and its north-west neighbour is also $3$. All other entries in the last diagonal are non-maximal, thus the statistic on this matrix is $2$. Conjecturally, this statistic is equidistributed with [[St000066]].

The number of inversions of the second entry of a permutation. This is, for a permutation $\pi$ of length $n$, $$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.