The number of B-inversions of a signed permutation. The number of B-inversions of a signed permutation $\sigma$ of length $n$ is $$\operatorname{inv}_B(\sigma) = \big|\{ 1 \leq i < j \leq n \mid \sigma(i) > \sigma(j) \}\big| + \big|\{ 1 \leq i \leq j \leq n \mid \sigma(-i) > \sigma(j) \}\big|,$$ see [1, Eq. (8.2)]. According to [1, Eq. (8.4)], this is the Coxeter length of $\sigma$.

The flag major index of a signed permutation. The flag major index of a signed permutation $\sigma$ is: $$\operatorname{fmaj}(\sigma)=\operatorname{neg}(\sigma)+2\cdot \sum_{i\in \operatorname{Des}_B(\sigma)}{i} ,$$ where $\operatorname{Des}_B(\sigma)$ is the $B$-descent set of $\sigma$; see [1, Eq.(10)]. This statistic is equidistributed with the $B$-inversions ([[St001428]]) and with the negative major index on the groups of signed permutations (see [1, Corollary 4.6]).

The flag Denert index of a signed permutation. The flag Denert index of a signed permutation $\sigma$ is: $$fden(\sigma) = \operatorname{neg}(\sigma) + 2 \cdot den_B(\sigma),$$ where $den_B(\sigma) = den(perm(\sigma))$ is the Denert index of the associated permutation to $\sigma$.

The sorting index of a signed permutation. A signed permutation $\sigma = [\sigma(1),\ldots,\sigma(n)]$ can be sorted $[1,\ldots,n]$ by signed transpositions in the following way: First move $\pm n$ to its position and swap the sign if needed, then $\pm (n-1), \pm (n-2)$ and so on. For example for $[2,-4,5,-1,-3]$ we have the swaps $$[2,-4,5,-1,-3] \rightarrow [2,-4,-3,-1,5] \rightarrow [2,1,-3,4,5] \rightarrow [2,1,3,4,5] \rightarrow [1,2,3,4,5]$$ given by the signed transpositions $(3,5), (-2,4), (-3,3), (1,2)$. If $(i_1,j_1),\ldots,(i_n,j_n)$ is the decomposition of $\sigma$ obtained this way (including trivial transpositions) then the sorting index of $\sigma$ is defined as $$\operatorname{sor}_B(\sigma) = \sum_{k=1}^{n-1} j_k - i_k - \chi(i_k < 0),$$ where $\chi(i_k < 0)$ is 1 if $i_k$ is negative and 0 otherwise. For $\sigma = [2,-4,5,-1,-3]$ we have $$\operatorname{sor}_B(\sigma) = (5-3) + (4-(-2)-1) + (3-(-3)-1) + (2-1) = 13.$$

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.