The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. In other words, it is the sum of the coefficients in $$(-1)^{|\lambda|-\ell(\lambda)}\nabla p_\lambda \vert_{q=1,t=1},$$ when expanded in the monomial basis. Here, $\nabla$ is the linear operator on symmetric functions where the modified Macdonald polynomials are eigenvectors. See the Sage documentation for definition and references [[http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/sfa.html]]

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 number of nonempty primitive factors of a binary word. A word $u$ is a factor of a word $w$ if $w = p u s$ for words $p$ and $s$. A word is primitive, if it is not of the form $u^k$ for a word $u$ and an integer $k\geq 2$. Apparently, the maximal number of nonempty primitive factors a binary word of length $n$ can have is given by [[oeis:A131673]].

The number of distinct subsequences in a binary word. In contrast to the subword complexity [[St000294]] this is the cardinality of the set of all subsequences of not necessarily consecutive letters.

The external path length of a binary tree. This is the sum of the lengths of all paths from the root to an external node, see Section 2.3.4.5 of [1]. This is also called the Sackin balance index of a rooted binary tree, see [2].

The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra.

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 rank of the permutation. This is its position among all permutations of the same size ordered lexicographically. This can be computed using the Lehmer code of a permutation: $$\text{rank}(\sigma) = 1 +\sum_{i=1}^{n-1} L(\sigma)_i (n − i)!,$$ where $L(\sigma)_i$ is the $i$-th entry of the Lehmer code of $\sigma$.

The sum of the entries of a semistandard tableau.

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]].