The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.

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 number of linear extensions of the tree. We use Knuth's hook length formula for trees [pg.70, 1]. For an ordered tree $T$ on $n$ vertices, the number of linear extensions is $$ \frac{n!}{\prod_{v\in T}|T_v|}, $$ where $T_v$ is the number of vertices of the subtree rooted at $v$.

The number of parking functions supported by a Dyck path. One representation of a parking function is as a pair consisting of a Dyck path and a permutation $\pi$ such that if $[a_0, a_1, \dots, a_{n-1}]$ is the area sequence of the Dyck path then the permutation $\pi$ satisfies $pi_i < pi_{i+1}$ whenever $a_{i} < a_{i+1}$. This statistic counts the number of permutations $\pi$ which satisfy this condition.

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