The number of invariant subsets of size 3 when acting with a permutation of given cycle type.

The number of distinct even parts of a partition. See Section 3.3.1 of [1].

Dyson's crank of a partition. Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as $$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$

The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].

The order dimension of the partition. Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.

The height of the bicoloured Motzkin path associated with the Dyck path.

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

The cardinality of a minimal cycle-isolating set of a graph. Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$. This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.

The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.