The inversion sum of a permutation. A pair $a < b$ is an inversion of a permutation $\pi$ if $\pi(a) > \pi(b)$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$. This is also half of the metric associated with Spearmans coefficient of association $\rho$, $\sum_i (\pi_i - i)^2$, see [5]. This is also equal to the total number of occurrences of the classical permutation patterns $[2,1], [2, 3, 1], [3, 1, 2]$, and $[3, 2, 1]$, see [2]. This is also equal to the rank of the permutation inside the alternating sign matrix lattice, see references [2] and [3]. This lattice is the MacNeille completion of the strong Bruhat order on the symmetric group [1], which means it is the smallest lattice containing the Bruhat order as a subposet. This is a distributive lattice, so the rank of each element is given by the cardinality of the associated order ideal. The rank is calculated by summing the entries of the corresponding ''monotone triangle'' and subtracting $\binom{n+2}{3}$, which is the sum of the entries of the monotone triangle corresponding to the identity permutation of $n$. This is also the number of bigrassmannian permutations (that is, permutations with exactly one left descent and one right descent) below a given permutation $\pi$ in Bruhat order, see Theorem 1 of [6].

Number of minimal entries in the Bruhat order matrix of a permutation. Associate to a permutation $\sigma$ of length $n$ the $n \times n$ matrix with entries $$r_{ij}(\sigma) = \left| \big\{ u \in \{1,\dots,i\} \mid \sigma(u) \leq j \big\}\right|.$$ For the identity permutation, one has $r_{ij} = \min\{i,j\}$, and $\sigma \leq \tau$ in the (strong) Bruhat order if and only if $r_{ij}(\tau) \leq r_{ij}(\sigma)$ for all $i,j$. This statistic records the number of indices $i,j$ with $r_{ij} = \min\{i,j\}$.

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 rank of the alternating sign matrix in the alternating sign matrix poset. This rank is the sum of the entries of the monotone triangle minus $\binom{n+2}{3}$, which is the smallest sum of the entries in the set of all monotone triangles with bottom row $1\dots n$. Alternatively, $rank(A)=\frac{1}{2} \sum_{i,j=1}^n (i-j)^2 a_{ij}$, see [3, thm.5.1].

The non-inversion sum of a permutation. A pair $a < b$ is an noninversion of a permutation $\pi$ if $\pi(a) < \pi(b)$. The non-inversion sum is given by $\sum(b-a)$ over all non-inversions of $\pi$.

The number of transpositions that are smaller or equal to a permutation in Bruhat order. A statistic is known to be '''smooth''' if and only if this number coincides with the number of inversions. This is also equivalent for a permutation to avoid the two pattern $4231$ and $3412$.

Lusztig's a-function for the symmetric group. Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$ by the Robinson-Schensted correspondence and $\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$ where $Q(x)'$ is the transposed tableau. Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$. See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.

The number of subgraphs. Given a graph $G$, this is the number of graphs $H$ such that $H \hookrightarrow G$.

The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.

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'''.