The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block and $j$ is a singleton block.

The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. This is the number of pairs $i\lt j$ in different blocks such that $i$ is a singleton block and $j$ is the maximal element of a block.

The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.

The Frankl number of a lattice. For a lattice $L$ on at least two elements, this is $$ \max_x(|L|-2|[x, 1]|), $$ where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.

The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].

The number of 2-regular simple modules in the incidence algebra of the lattice.