The cosine of a permutation. For a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this is given by $\sum_{i=1}^n (i\pi_i)$. The name comes from the observation that this equals $\frac{n(n+1)(2n+1)}{6}\cos(\theta)$ where $\theta$ is the angle between the vector $(\pi_1,\ldots,\pi_n)$ and the vector $(1,\ldots,n)$, see [1].

The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.

The sum of the entries of the Gelfand-Tsetlin pattern.

The sum of the major and the inverse major index of a permutation. This statistic is the sum of [[St000004]] and [[St000305]].

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 load of a permutation. The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.

The number of tolerances of a finite lattice. Let $L$ be a lattice. A tolerance $\tau$ is a reflexive and symmetric relation on $L$ which is compatible with meet and join. Equivalently, a tolerance of $L$ is the image of a congruence by a surjective lattice homomorphism onto $L$. The number of tolerances of a chain of $n$ elements is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$, see [2].