The global dimension of the incidence algebra of the lattice over the rational numbers.

The number of rises of length 3 of a Dyck path.

The multiplicity of the sign representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.

The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.

The number of symmetric hooks on the diagonal of a partition.

The number of invariant subsets of size 3 when acting with a permutation of given cycle type.

The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.

The number of loops of the quiver corresponding to the reduced incidence algebra of a poset.

The number of non-isomorphic posets with precisely one further covering relation.

The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.