The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. Precisely the 132-avoiding permutations have only one associated reduced pipe dream.

The sum of the positions of the strong records of an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. This statistic is the sum of the positions of the strong records.

The major index of a permutation. This is the sum of the positions of its descents, $$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$ Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$. A statistic equidistributed with the major index is called '''Mahonian statistic'''.

The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].

The sum of the descent bottoms of a permutation. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$ For the descent tops, see [[St000111]].

The Denert index of a permutation. It is defined as $$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$ where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].

The inverse major index of a permutation. This is the major index [[St000004]] of the inverse permutation [[Mp00066]].

The maz index, the major index of a permutation after replacing fixed points by zeros. The descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].

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 $w_i < w_l$.

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.