The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.

The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

The number of admissible inversions of a permutation. Let $w = w_1,w_2,\dots,w_k$ be a word of length $k$ with distinct letters from $[n]$. An admissible inversion of $w$ is a pair $(w_i,w_j)$ such that $1\leq i < j\leq k$ and $w_i > w_j$ that satisfies either of the following conditions: $1 < i$ and $w_{i−1} < w_i$ or there is some $l$ such that $i < l < j$ and $w_i < w_l$.

The minimal number of transpositions needed to sort a permutation in either direction. For a permutation $\sigma$, this is $\min\{ \operatorname{inv}(\sigma),\operatorname{inv}(\tau)\}$ where $\tau$ is the reverse permutation sending $i$ to $\sigma(n+1-i)$.

The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by $$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$

The number of inversions of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. A pair $(i,j)$ is an inversion of the word $w$ if $w_i > w_j$.

The number of occurrences of the vincular pattern |132 in a permutation. This is the number of occurrences of the pattern $(1,3,2)$, such that the letter matched by $1$ is the first entry of the permutation.

The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). In symbols, for a permutation $\pi$ this is $$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 1 \leq i_1,\ldots,i_k \leq n\},$$ where $\tau_a = (a,a+1)$ for $1 \leq a \leq n$ and $n+1$ is identified with $1$. Put differently, this is the number of cyclically simple transpositions needed to sort a permutation.

The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.