The number of cyclic misalignments of a permutation. The pair $(i,j)$ is a cyclic alignment of a permutation $\pi$ if $i, \pi(i), j, \pi(j)$ are cyclically ordered and all distinct, see Section 5 of [1].

The global dimension of the incidence algebra of the lattice over the rational numbers.

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.

The number of potential covers of a poset. A potential cover is a pair of uncomparable elements $(x, y)$ which can be added to the poset without adding any other relations. For example, let $P$ be the disjoint union of a single relation $(1, 2)$ with the one element poset $0$. Then the relation $(0, 1)$ cannot be added without adding also $(0, 2)$, however, the relations $(0, 2)$ and $(1, 0)$ are potential covers.

The permanent of the Coxeter matrix of the poset.

The number of posets with the same order polynomial. The order polynomial of a poset $P$ is the polynomial $S$ such that $S(m)$ is the number of order-preserving maps from $P$ to $\{1,\dots,m\}$. See sections 3.12 and 3.15 of [1].