The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.

The number of 2-regular simple modules in the incidence algebra of the lattice.

The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of $GL_2$.

The Colin de Verdière graph invariant.

The diagonal index (content) of a partition. The '''diagonal index''' of the cell at row $r$ and column $c$ of a partition is $c - r$; this is sometimes called the '''content''' of the cell. The '''diagonal index of a partition''' is the sum of the diagonal index of each cell of the partition.

The Plancherel distribution on integer partitions. This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions. Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.

Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1] $$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$ equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.

The number of invariant oriented cycles when acting with a permutation of given cycle type.

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$

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]].