The leading coefficient of the rook polynomial of an integer partition. Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.

The length of the longest weakly inreasing subsequence of parts of an integer composition.

The number of weak records in an integer composition. A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.

The maximal number of repetitions of an integer composition.

The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

The dominant dimension of the corresponding Comp-Nakayama algebra.

The number of ways to place as many non-attacking rooks as possible on a Ferrers board.

The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. Such a partition always exists because of a construction due to Dudek and Pralat [1] and independently Pokrovskiy [2].

The number of weak descents in an integer composition. A weak descent of an integer composition $\alpha=(a_1, \dots, a_n)$ is an index $1\leq i < n$ such that $a_i \geq a_{i+1}$.

The number of distinct eigenvalues of the distance Laplacian of a connected graph.