The number of inversions of a standard tableau. Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

The breadth of a lattice. The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.

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 parts of an integer partition that are at least two.

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 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 $k$.