The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

The Grundy value for the game of deleting vertices of a graph until it has no edges.

The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k$$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.

The number of pixed points of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$\textrm{pix} \, \sigma = \textrm{length} (p)$$ .