The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra.

The dimension of the maximal parabolic seaweed algebra corresponding to the partition. Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be two compositions of $n$. The corresponding seaweed algebra is the associative subalgebra of the algebra of $n\times n$ matrices which preserves the flags $$ \{0\} \subset V_1 \subset \cdots \subset V_{m-1} \subset V_m =V $$ and $$ V=W_0\supset W_1\supset \cdots \supset W_t=\{0\}, $$ where $V_i=\text{span}\{e_1,\dots, e_{a_1+\cdots +a_i}\}$ and $W_j=\text{span}\{e_{b_1+\cdots +b_j+1},\dots, e_n\}$. Thus, its dimension is $$ \frac{1}{2}\left(\sum a_i^2 + \sum b_i^2\right). $$ It is maximal parabolic if $b_1=n$.

The number of coloured endofunctions such that the multiplicities of colours are given by a partition. In particular, the value on the partition $(n)$ is the number of endofunctions on $n$ vertices up to relabelling, [[oeis:A000088]], whereas the value on the partition $(1^n)$ is the number of endofunctions [[oeis:A000312]].

The rank of the permutation. This is its position among all permutations of the same size ordered lexicographically. This can be computed using the Lehmer code of a permutation: $$\text{rank}(\sigma) = 1 +\sum_{i=1}^{n-1} L(\sigma)_i (n − i)!,$$ where $L(\sigma)_i$ is the $i$-th entry of the Lehmer code of $\sigma$.

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.

The number of Dyck paths that are weakly below a Dyck path.

Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1.

The composition number of a graph. This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1).

The number of tolerances of a finite lattice. Let $L$ be a lattice. A tolerance $\tau$ is a reflexive and symmetric relation on $L$ which is compatible with meet and join. Equivalently, a tolerance of $L$ is the image of a congruence by a surjective lattice homomorphism onto $L$. The number of tolerances of a chain of $n$ elements is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$, see [2].