The number of right floats of a permutation. Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of $\pi$. See Definition 3.10 and Example 3.11 in [1].

Number of triples of incomparable elements in a finite poset. For a finite poset this is the number of 3-element sets $S \in \binom{P}{3}$ that are pairwise incomparable.

The leading coefficient of the Poincare polynomial of the poset cone. For a poset $P$ on $\{1,\dots,n\}$, let $\mathcal K_P = \{\vec x\in\mathbb R^n| x_i < x_j \text{ for } i < _P j\}$. Furthermore let $\mathcal L(\mathcal A)$ be the intersection lattice of the braid arrangement $A_{n-1}$ and let $\mathcal L^{int} = \{ X \in \mathcal L(\mathcal A) | X \cap \mathcal K_P \neq \emptyset \}$. Then the Poincare polynomial of the poset cone is $Poin(t) = \sum_{X\in\mathcal L^{int}} |\mu(0, X)| t^{codim X}$. This statistic records its leading coefficient.

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

The number of connected components of the Hasse diagram for the poset.

The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.

The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.