The number of non-empty boolean intervals in a poset.

The sum of the vertex degrees of a graph. This is clearly equal to twice the number of edges, and, incidentally, also equal to the trace of the Laplacian matrix of a graph. From this it follows that it is also the sum of the squares of the eigenvalues of the adjacency matrix of the graph. The Laplacian matrix is defined as $D-A$ where $D$ is the degree matrix (the diagonal matrix with the vertex degrees on the diagonal) and where $A$ is the adjacency matrix. See [1] for detailed definitions.

The Szeged index minus the Wiener index of a graph. It is known that the Szeged index is at least as much as the Wiener index. For $2$-connected graphs on $n$ vertices, the difference is at least $2n-6$. For the two individual statistics see [[St000263]] and [[St000265]].

The size of the centralizer of any permutation of given cycle type. The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$: $$C_g = \{h \in G : hgh^{-1} = g\}.$$ Its size thus depends only on the conjugacy class of $g$. The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is $$|C| = \Pi j^{a_j} a_j!$$ For example, for any permutation with cycle type $\lambda = (3,2,2,1)$, $$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$ There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.

The leading coefficient of the rook polynomial of an integer partition. Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.

The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to $$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$ for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].

The product of the parts of an integer partition.

The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.

The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that $$ \alpha\pi\alpha^{-1} = \pi^{-1}.$$

The weight of a partition according to Alladi.