The number of alternating sign matrices whose left key is the permutation. The left key of an alternating sign matrix was defined by Lascoux in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.

The number of semistandard tableaux on a given integer partition with minimal maximal entry. This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$. Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly, $$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$ where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell. See [Theorem 6.3, 1] for details.

The number of reduced Kogan faces with the permutation as type. This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

The number of minimal elements in a poset.

The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.