The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs.

The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \}$ of $\{1,2,3 \}$ is decoded as 101.

The number of cyclical small weak excedances. A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.

The number of small weak excedances. A small weak excedance is an index $i$ such that $\pi_i \in \{i,i+1\}$. This is the sum of [[St000022]] and [[St000237]].

The number of cyclical small excedances. A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.

The rix statistic of a permutation. This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then $rix(w) := 0$ if $i = 1 < k$, $rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and $rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.

The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block.

The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block.

The number of right floats of a permutation. Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of $\pi$. See Definition 3.10 and Example 3.11 in [1].

The aix statistic of a permutation. According to [1], this statistic on finite strings $\pi$ of integers is given as follows: let $m$ be the leftmost occurrence of the minimal entry and let $\pi = \alpha\ m\ \beta$. Then $$\operatorname{aix}\pi = \begin{cases} \operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\ 1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\ 0 & \text{ if } \beta = \emptyset \end{cases}\ .$$