The number of reduced words for a permutation. This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.

The number of [[/StandardTableaux|standard Young tableaux]] of the partition.

The number of standard Young tableaux of the skew partition.

The number of reduced words of a signed permutation. This is the number of ways to write a permutation as a minimal length product of simple reflections.

The number of linear extensions of a poset.

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1]. This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for vertex labelled trees, whose reversal of the generating function $f^{(-1)}(x) = x\exp(-x)$, see [1, sec. 3.3] Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. Then this statistic gives the number of ways of putting a rooted tree on this set of colored vertices so that no leaf is the same color as its parent.

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

The weight of a partition according to Alladi.

The alternating sum of the coefficients of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1].

The number of crossings plus two-nestings of a perfect matching. This is $C+2N$ where $C$ is the number of crossings ([[St000042]]) and $N$ is the number of nestings ([[St000041]]). The generating series $\sum_{m} q^{\textrm{cn}(m)}$, where the sum is over the perfect matchings of $2n$ and $\textrm{cn}(m)$ is this statistic is $[2n-1]_q[2n-3]_q\cdots [3]_q[1]_q$ where $[m]_q = 1+q+q^2+\cdots + q^{m-1}$ [1, Equation (5,4)].