The product of the factorials of the multiplicities of an integer partition.

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.

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 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].