The maximal number of nonzero entries on a diagonal of a permutation matrix. For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is $$\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix},$$ and the entries corresponding to $\pi_2=1$, $\pi_3=2$ and $\pi_5=4$ are all on the fourth diagonal from the right. In other words, this is $\max_k \lvert\{i: \pi_i-i = k\}\rvert$

The maximal sum of entries on a diagonal of an alternating sign matrix. For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)$$ are $(0,1,1,0,1,1,0)$, so the statistic is $1$. This is a natural extension of [[St000887]] to alternating sign matrices.

The maximal number of nonzero entries on a diagonal of an alternating sign matrix. For example, for the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)$$ the numbers of nonzero entries are $(0,1,1,2,1,1,0)$, so the statistic is $2$. This is a natural extension of [[St000887]] to alternating sign matrices. See [[St000888]] for the maximal sums.