The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.

Half the size of the symmetry class of a permutation. The symmetry class of a permutation $\pi$ is the set of all permutations that can be obtained from $\pi$ by the three elementary operations '''inverse''' ([[Mp00066]]), '''reverse''' ([[Mp00064]]), and '''complement''' ([[Mp00069]]). This statistic is undefined for the unique permutation on one element, because its value would be $1/2$.

The length of the longest pattern of the form k 1 2...(k-1).

The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.

The length of the longest cycle of a permutation.

The number of cyclic descents of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.

The number of parts of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of parts of the shifted shape.

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$

The orbit size of a permutation under Foata's bijection.

The number of minimal length factorizations of a permutation into star transpositions. For a permutation $\pi\in\mathfrak S_n$ a minimal length factorization into star transpositions is a factorization of the form $$\pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n,$$ where $\tau_a = (1,a)$ for $2 \leq a \leq n$ and $k$ is minimal. [1, lem.2.1] shows that the minimal length of such a factorization is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$, see [[St001077]]. [2, cor.2] shows that the number of such minimal factorizations is $$ \frac{(n+m-2(k+1))!}{(n-k)!}\ell_1\cdots\ell_m, $$ where $\ell_1,\dots,\ell_m$ is the cycle type of $\pi$ and $k$ is the number of fixed point different from $1$.