The positive inversions of an alternating sign matrix. This is defined as $$\sum_{i > k,j < l} A_{ij}A_{kl} - \text{the number of negative ones in the matrix}.$$ After counter-clockwise rotation, this is also the number of osculations in the corresponding fan of Dyck paths.

The index of the step at the first peak of maximal height in a Dyck path.

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 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 flush statistic of a semistandard tableau. Let $T$ be a tableaux with $r$ rows such that each row is longer than the row beneath it by at least one box. Let $1 \leq i < k \leq r+1$ and suppose $l$ is the smallest integer greater than $k$ such that there exists an $l$-segment in the $(i+1)$-st row of $T$. A $k$-segment in the $i$-th row of $T$ is called '''flush''' if the leftmost box in the $k$-segment and the leftmost box of the $l$-segment are in the same column of $T$. If, however, no such $l$ exists, then this $k$-segment is said to be flush if the number of boxes in the $k$-segment is equal to difference of the number of boxes between the $i$-th row and $(i+1)$-st row. The flush statistic is given by the number of $k$-segments in $T$.

The sorting index of a permutation. The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols. Compare this to [[St000018]], the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.

The maf index of a permutation. Let $\sigma$ be a permutation with fixed point set $\operatorname{FIX}(\sigma)$, and let $\operatorname{Der}(\sigma)$ be the derangement obtained from $\sigma$ by removing the fixed points. Then $$\operatorname{maf}(\sigma) = \sum_{i \in \operatorname{FIX}(\sigma)} i - \binom{|\operatorname{FIX}(\sigma)|+1}{2} + \operatorname{maj}(\operatorname{Der}(\sigma)),$$ where $\operatorname{maj}(\operatorname{Der}(\sigma))$ is the major index of the derangement of $\sigma$.

The disorder of a permutation. Consider a permutation $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The '''disorder''' of $\pi$ is defined to be the number of times a position was not removed in this process. For example, the disorder of $[3,5,2,1,4]$ is $8$ since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.

The aid statistic in the sense of Shareshian-Wachs. This is the number of admissible inversions [[St000866]] plus the number of descents [[St000021]]. This statistic was introduced by John Shareshian and Michelle L. Wachs in [1]. Theorem 4.1 states that the aid statistic together with the descent statistic is Euler-Mahonian.

The minimal length of a factorization of a permutation using the permutations (12)(34)..., (23)(45)..., and (12). In symbols, for a permutation $\pi$ this is $$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k} \},$$ where, with $m_1$ the largest even number at most $n$ and $m_2$ the largest odd number at most $n$, each factor $\tau_i$ is one of the three permutations $(1,2)(3,4)\cdots(m_1-1,m_1)$ or $(2,3)(4,5)\cdots(m_2-1,m_2)$ or $(1,2)$.