The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.

The number of cycles in the breakpoint graph of a permutation. The breakpoint graph of a permutation $\pi_1,\dots,\pi_n$ is the directed, bicoloured graph with vertices $0,\dots,n$, a grey edge from $i$ to $i+1$ and a black edge from $\pi_i$ to $\pi_{i-1}$ for $0\leq i\leq n$, all indices taken modulo $n+1$. This graph decomposes into alternating cycles, which this statistic counts. The distribution of this statistic on permutations of $n-1$ is, according to [cor.1, 5] and [eq.6, 6], given by $$ \frac{1}{n(n+1)}((q+n)_{n+1}-(q)_{n+1}), $$ where $(x)_n=x(x-1)\dots(x-n+1)$.

The maximal multiplicity of an eigenvalue in a graph.

The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.

The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. In other words, this is the number of valleys and peaks whose first step is in odd position, the initial position equal to 1. The generating function is given in [1].

The number of vertices with out-degree 1 in a binary tree. See the references for several connections of this statistic. In particular, the number $T(n,k)$ of binary trees with $n$ vertices and $k$ out-degree $1$ vertices is given by $T(n,k) = 0$ for $n-k$ odd and $$T(n,k)=\frac{2^k}{n+1}\binom{n+1}{k}\binom{n+1-k}{(n-k)/2}$$ for $n-k$ is even.

The number of ascent tops in the flattened set partition such that all smaller elements appear before. Let $P$ be a set partition. The flattened set partition is the permutation obtained by sorting the set of blocks of $P$ according to their minimal element and the elements in each block in increasing order. Given a set partition $P$, this statistic is the binary logarithm of the number of set partitions that flatten to the same permutation as $P$.

The diameter of a connected graph. This is the greatest distance between any pair of vertices.