The value of the power-sum symmetric function evaluated at 1. The statistic is $p_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k$, where $\lambda$ has $k$ parts.

The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].

The number of induced subgraphs. A subgraph $H \subseteq G$ is induced if $E(H)$ consists of all edges in $E(G)$ that connect the vertices of $H$.

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.

The number of spanning subgraphs of a graph with the same connected components. A subgraph or factor of a graph is spanning, if it has the same vertex set [1]. The present statistic additionally requires the subgraph to have the same components. It can be obtained by evaluating the Tutte polynomial at the points $x=1$ and $y=2$, see [2,3]. By mistake, [2] refers to this statistic as the number of spanning subgraphs, which would be $2^m$, where $m$ is the number of edges. Equivalently, this would be the evaluation of the Tutte polynomial at $x=y=2$.

The product of the factorials of the parts.

The product of the parts of an integer partition.

The number of semistandard Young tableau of given shape, with entries at most 4. This is also the dimension of the corresponding irreducible representation of $GL_4$.

The clique-coclique number of a graph. This is the product of the size of a maximal clique [[St000097]] and the size of a maximal independent set [[St000093]].

The number of multipartitions of sizes given by an integer partition. This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.