The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. A graph is a split graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge and $(b,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

The multinomial of the parts of a partition. Given an integer partition $\lambda = [\lambda_1,\ldots,\lambda_k]$, this is the multinomial $$\binom{|\lambda|}{\lambda_1,\ldots,\lambda_k}.$$ For any integer composition $\mu$ that is a rearrangement of $\lambda$, this is the number of ordered set partitions whose list of block sizes is $\mu$.

The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.

The number of coarsenings of a partition. A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.

The multiplicity of the largest eigenvalue in a graph.

The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. For example, $e_{22} = s_{1111} + s_{211} + s_{22}$, so the statistic on the partition $22$ is $3$.

The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. The total number of words with letter multiplicities given by an integer partition is [[St000048]]. For example, there are twelve words with letters $0,0,1,2$ corresponding to the partition $[2,1,1]$. Two of these contain an increasing factor of length three: $0012$ and $0120$. Note that prescribing the multiplicities for different letters yields the same number. For example, there are also two words with letters $0,1,1,2$ containing an increasing factor of length three: $1012$ and $0121$. The number of words of length $n$ with letters in an alphabet of size $k$ avoiding the consecutive pattern $123$ is determined in [1].

The number of graphs with the same chromatic polynomial.

Number of standard Young tableaux of the skew shape tracing the border of the given partition. Let $\lambda \vdash n$ be a diagram with the given partition as shape. Add $n$ additional boxes, one in each column $1,\dotsc,n$, and let this be $\mu$. The statistic is the number of standard Young tableaux of skew shape $\mu/\lambda$, which is equal to $\frac{n!}{\prod_{i} (\mu_i - \lambda_i)!}$. For example, $\lambda=[2,1,1]$ gives $\mu = [4,2,1,1]$. The first row in the skew shape $\mu/\lambda$ has two boxes, so the number of SYT of shape $\mu/\lambda$ is then $4!/2 = 12$. This statistic shows up in the study of skew specialized Macdonald polynomials, where a type of charge statistic give rise to a $q$-analogue of the above formula.

The induced matching number of a graph. An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.