The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$

The register function (or Horton-Strahler number) of a binary tree. This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.

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 constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.

This is the number of standard Young tableaux of the given shifted shape. For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e., $$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$ In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.