The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. This is also the number of '''twin vertices''' in the unit interval graph corresponding to the Dyck path. A Dyck path of size $n$ determines a unit interval graph on vertex set $[n]$, see [1]. Two vertices ${i,i+1}$ are '''twins''' if they have the same closed neighbourhood in the unit interval graph. Put differently, this is the number of indices $i$ such that the $i$-th and $(i+1)$-st up steps are consecutive, and the $i$-th and $(i+1)$-st down steps are consecutive.

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.

The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).

The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.