The biggest entry in the block containing the 1.

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 column of the unique '1' in the first row of the alternating sign matrix. The generating function of this statistic is given by $$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$ see [2].

The last entry of a permutation. This statistic is undefined for the empty permutation.

The size of the left subtree of a binary tree.

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.

The position of the first return of a Dyck path.

The row of the unique '1' in the first column of the alternating sign matrix.

The column of the unique '1' in the last row of the alternating sign matrix.

The row of the unique '1' in the last column of the alternating sign matrix.