The cardinality of the support of a permutation. A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$. The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$. Thus, the connectivity set is the complement of the support.

The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.

The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.

The inversion number of the alternating sign matrix. If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as $$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$ When restricted to permutation matrices, this gives the usual inversion number of the permutation.

The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

The number of indices that are not small excedances. A small excedance is an index $i$ for which $\pi_i = i+1$.

The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.

The length of the partition.

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].