The number of maximal chains in a poset.

The largest part of an integer partition.

The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.

The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.

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 number of compositions obtained by rotating the composition.

The number of maximal chains of maximal size in a poset.

The number of ordered refinements of an integer partition. This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.

The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$