The excess length of a longest path consisting of elements and blocks of a set partition. Let $p$ be a set partition of $\{1,\dots,n\}$. Let $G$ be the graph with edges $(i,i+1)$ for $i\in\{1,\dots,n-1\}$ and $(i, b)$, whenever $i$ is an element of a non-singleton block $b\in p$. Then this statistic records the length of the longest path from $1$ to $n$ in $G$, reduced by $n$. Conjecturally, a longest path has more than $n$ vertices provided that the set partition has no singletons.

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

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.

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.