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 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].

The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.

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 atomic length of a signed permutation. The atomic length of an element $w$ of a Weyl group is the sum of the heights of the inversions of $w$.

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

The sum of the entries reduced by the index of their row in a semistandard tableau. This is also the depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$, independent of the Cartan rank.

The number of strict 3-descents. A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $i+3 \leq n$ and $\pi(i) > \pi(i+3)$.

The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.

The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$. Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].