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 $\lambda$.

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 number of factors of a lattice as a Cartesian product of lattices. Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.

The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

The row sums of the character table of the symmetric group. Equivalently, this is the multiplicity of the irreducible representation corresponding to the given partition in the adjoint representation of the symmetric group.

The number of integer partitions of n that are dominated by an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.

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

The number of ordered refinements of an integer partition. This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.

The number of simple reflexive modules in the corresponding Nakayama algebra.