The length of the longest partition in the vacillating tableau corresponding to a set partition. To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$. Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row. Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$. This statistic is the length of the longest partition on the diagonal of the diagram.

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

The nesting number of a set partition. This is the maximal number of chords in the standard representation of a set partition that mutually nest.

The crossing number of a set partition. This is the maximal number of chords in the standard representation of a set partition, that mutually cross.

The length of the maximal rise of a Dyck path.

The number of nonsingleton blocks of a set partition.

The maximal area to the right of an up step of a Dyck path.

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.