For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). The statistic gives half of the rank of the matrix C^t-C.

The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. Two pairs $(a,b)$ and $(c,d)$ (with $a < b$ and $c < d$) in a perfect matching cross if and only if $a < c < b < d$ or $c < a < d < b$. Define a skew symmetric matrix $M$ whose rows and columns are indexed by the pairs of the matching, with $$ M_{(a,b),(c,d)} = \begin{cases} 1 &\text{if \(a < c < b < d\)}\\ -1 &\text{if \(c < a < d < b\)}\\ 0 &\text{otherwise} \end{cases} $$ The rank of this matrix is always even. The present statistic is half of the matrix' rank.

The genus of a permutation. The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation $$ n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ), $$ where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.

The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.