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 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 vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra.

The inversion index of a permutation. The ''inversion index'' of a permutation $\sigma=\sigma_1\sigma_2\ldots\sigma_n$ is defined as $$ \sum_{\mbox{inversion pairs } (\sigma_i,\sigma_j)} \sigma_i $$ where $(\sigma_i,\sigma_j)$ is an inversion pair if i < j and $\sigma_i > \sigma_j$. This equals the sum of the entries in the corresponding descending plane partition; see [1].

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 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 index of a given binary word in the lex-order among all its cyclic shifts.

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 major index of a Dyck path. This is the sum over all $i+j$ for which $(i,j)$ is a valley of $D$. The generating function of the major index yields '''MacMahon''' 's $q$-Catalan numbers $$\sum_{D \in \mathfrak{D}_n} q^{\operatorname{maj}(D)} = \frac{1}{[n+1]_q}\begin{bmatrix} 2n \\ n \end{bmatrix}_q,$$ where $[k]_q := 1+q+\ldots+q^{k-1}$ is the usual $q$-extension of the integer $k$, $[k]_q!:= [1]_q[2]_q \cdots [k]_q$ is the $q$-factorial of $k$ and $\left[\begin{smallmatrix} k \\ l \end{smallmatrix}\right]_q:=[k]_q!/[l]_q![k-l]_q!$ is the $q$-binomial coefficient. The major index was first studied by P.A.MacMahon in [1], where he proved this generating function identity. There is a bijection $\psi$ between Dyck paths and '''noncrossing permutations''' which simultaneously sends the area of a Dyck path [[St000012]] to the number of inversions [[St000018]], and the major index of the Dyck path to $n(n-1)$ minus the sum of the major index and the major index of the inverse [2]. For the major index on other collections, see [[St000004]] for permutations and [[St000290]] for binary words.