The number of alignments of a signed permutation. An alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $-n \leq i \leq j \leq n$, $i,j\neq 0$, such that one of the following conditions hold: * $i < j \leq \pi(j) < \pi(i)$, and $j > 0$ if it is a fixed point, or * $\pi(j) < \pi(i) \leq i < j)$ and $i < 0$ if it is a fixed point, or * $i \leq \pi(i) < \pi(j) \leq j$ and $i > 0$ if it is a fixed point and $j < 0$ if it is a fixed point, or * $\pi(i) \leq i < j \leq \pi(j)$ and $i < 0$ if it is a fixed point and $j > 0$ if it is a fixed point. Let $al$ be the number of alignments of $\pi$, $cr$ be the number of crossings, [[St001862]], and let $fwex$ be the number of flag weak excedances, [[St001817]]. Then $$2 cr + al = n^2 - 2n + fwex.$$

The dimension of the irreducible representation of Sp(4) labelled by an integer partition. Consider the symplectic group $Sp(2n)$. Then the integer partition $(\mu_1,\dots,\mu_k)$ of length at most $n$ corresponds to the weight vector $(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$. For example, the integer partition $(2)$ labels the symmetric square of the vector representation, whereas the integer partition $(1,1)$ labels the second fundamental representation.