Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition.

The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. Let $\pi$ be a permutation of cycle type $\mu$. A transitive monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ is a tuple of $r = n + \ell(\mu) - 2$ transpositions $$ (a_1, b_1),\dots,(a_r, b_r) $$ with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, such that the subgroup of $\mathfrak S_n$ generated by the transpositions acts transitively on $\{1,\dots,n\}$ and hose product, in this order, is $\pi$.

Twelve times the variance of the major index among all standard Young tableaux of a partition. For a partition $\lambda$ of $n$, this variance is given in [1, Proposition 3.2] by $$\frac{1}{12}\Big(\sum_{k = 1}^n i^2 - \sum_{i,j \in \lambda} h_{ij}^2\Big),$$ where the second sum ranges over all cells in $\lambda$ and $h_{ij}$ is the hook length of the cell $(i,j) \in \lambda$.

The number of coloured connected graphs such that the multiplicities of colours are given by a partition. In particular, the value on the partition $(n)$ is the number of unlabelled connected graphs on $n$ vertices, [[oeis:A001349]], whereas the value on the partition $(1^n)$ is the number of labelled connected graphs [[oeis:A001187]].

The number of transitive factorisations of a permutation of given cycle type into star transpositions. Let $\pi$ be a permutation of cycle type $\lambda\vdash n$ and let $r=n + \ell(\lambda) - 2$. A minimal factorization of $\pi$ into star transpositions is an $r$-tuple of transpositions $(1, a_1)\dots(1, a_r)$ whose product (in this order) equals $\pi$. The number of such factorizations equals [1] $$ \frac{r!}{n!} \lambda_1\dots\lambda_{\ell(\lambda)}. $$

Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path).

The number of Dyck paths that are weakly below a Dyck path, except for the path itself.

The number of occurrences of the pattern 123 or of the pattern 312 in a permutation.

The number of strictly order preserving maps of a poset into itself. A map $f$ is strictly order preserving if $a < b$ implies $f(a) < f(b)$.

Babson and Steingrímsson's statistic of a permutation. In terms of generalized patterns this is $$ (13-2) + (21-3) + (32-1) + (21). $$ Here, $(\pi)$ denotes the number of times the pattern $\pi$ occurs in a permutation, and letters in the pattern which are not separated by a dash must appear consecutively.