The size of the largest ordinal summand in the poset. The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$. This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.

The largest part of an integer partition.

The largest part of an integer composition.

Maximum difference of elements in cycles. Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$. The statistic is then the maximum of this value over all cycles in the permutation.

The length of a longest path in a graph.

The length of the partition.

The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].

The last entry in the first row of a standard tableau.

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".