The number of cycles in the breakpoint graph of a permutation. The breakpoint graph of a permutation $\pi_1,\dots,\pi_n$ is the directed, bicoloured graph with vertices $0,\dots,n$, a grey edge from $i$ to $i+1$ and a black edge from $\pi_i$ to $\pi_{i-1}$ for $0\leq i\leq n$, all indices taken modulo $n+1$. This graph decomposes into alternating cycles, which this statistic counts. The distribution of this statistic on permutations of $n-1$ is, according to [cor.1, 5] and [eq.6, 6], given by $$ \frac{1}{n(n+1)}((q+n)_{n+1}-(q)_{n+1}), $$ where $(x)_n=x(x-1)\dots(x-n+1)$.

Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.

Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$.

Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.

The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. In other words, this is $2^k$ where $k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition. The generating function for the number of equivalence classes, $f(n)$, is $$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$$

The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. Equivalently, given an integer partition $\lambda$, this is the number of molecular combinatorial species that decompose into a product of atomic species of sizes $\lambda_1,\lambda_2,\dots$. In particular, the value on the partition $(n)$ is the number of atomic species of degree $n$, [2].

The number of orbits of promotion on a graph. Let $(V, E)$ be a graph with $n=|V|$ vertices, and let $\sigma: V \to [n]$ be a labelling of its vertices. Let $ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $\{\sigma^{-1}(i), \sigma^{-1}(j)\}\in E$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $ The promotion operator is the product $\tau_{n-1,n}\dots\tau_{1,2}$. This statistic records the number of orbits in the orbit decomposition of promotion.

The number of monotone factorisations of genus zero of a permutation of given cycle type. A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ 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 $i$, whose product, in this order, is $\pi$. For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.

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] $$ \frac{r!}{n!} \lambda_1\dots\lambda_{\ell(\lambda)}. $$