The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. This is the multinomial of the multiplicities of the parts, see [1]. This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$, where $k$ is the number of parts of $\lambda$. An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$ where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.

Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.

The number of crossing-similar perfect matchings of a perfect matching. Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$. Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of crossings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$. [thm 1.2, 1] shows that to find out whether $M$ and $N$ are crossing-similar, it is enough to check that $M$ and $N$ have the same number of crossings, and that the distribution of crossings agrees for their direct children. [thm 3.3, 1], see also [2], gives the number of equivalence classes of crossing-similar matchings with $n$ arcs as $$2^{n-2}\left(\binom{n}{2}+2\right).$$

The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks. This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word $0110$: $$ 0110 < 1011 < 1101 < 1110 < 1111 $$ and $$ 0110 < 1010 < 1101 < 1110 < 1111. $$

The number of alignments in the permutation.

The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.