The Edelman-Greene number of a permutation. This is the sum of the coefficients of the expansion of the Stanley symmetric function $F_\omega$ in Schur functions. Equivalently, this is the number of semistandard tableaux whose column words - obtained by reading up columns starting with the leftmost - are reduced words for $\omega$.

The number of non-isomorphic sublattices of a lattice.

The number of non-isomorphic subposets of a lattice which are lattices.

The number of standard Young tableaux whose descent set is the binary word. A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation. For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$.

The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

The number of graphs with the same chromatic polynomial.

The number of graphs with the same symmetric edge polytope as the given graph. The symmetric edge polytope of a graph on $n$ vertices is the polytope in $\mathbb R^n$ defined as the convex hull of $e_i - e_j$ and $e_j - e_i$ for each edge $(i, j)$, where $e_1,\dots, e_n$ denotes the standard basis.

The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. A graph is a split graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge and $(b,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. A graph is outerplanar if and only if in any linear ordering of its vertices, there are no four vertices $a < b < c < d$ such that $(a,c)$ and $(b,d)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

The monophonic position number of a graph. A subset $M$ of the vertex set of a graph is a monophonic position set if no three vertices of $M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.