The number of parts of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of parts of the shifted shape.

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

The Edelman-Greene number of a permutation. This is the sum of the coefficients of the expansion of the Stanley symmetric function $F_\omega$ in Schur functions. Equivalently, this is the number of semistandard tableaux whose column words - obtained by reading up columns starting with the leftmost - are reduced words for $\omega$.

The length of the shortest maximal antichain in a poset.

The number of connected components of a skew partition.

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.