The number of two-component spanning forests of a graph. A '''spanning subgraph''' is a subgraph which contains all vertices of the ambient graph. A '''forest''' is a graph which contains no cycles, and has any number of connected components. A '''two-component spanning forest''' is a spanning subgraph which contains no cycles and has two connected components.

The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.

The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.

The number of characters of the symmetric group whose value on the partition is positive.

The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].

The Ramsey number of a graph. This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1] Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]

The order of toric promotion on the set of labellings of a graph. In the context of toric promotion, a labelling of a graph $(V, E)$ with $n=|V|$ vertices is a bijection $\sigma: V \to [n]$. In particular, any graph has $n!$ labellings.

The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.

The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.