The number of triconnected components of a graph. A connected graph is '''triconnected''' or '''3-vertex connected''' if it cannot be disconnected by removing two or fewer vertices. An arbitrary connected graph can be decomposed as a union of biconnected (2-vertex connected) graphs, known as '''blocks''', and each biconnected graph can be decomposed as a union of components with are either a cycle (type "S"), a cocyle (type "P"), or triconnected (type "R"). The decomposition of a biconnected graph into these components is known as the '''SPQR-tree''' of the graph.

The genus of a graph. This is the smallest genus of an oriented surface on which the graph can be embedded without crossings. One can indeed compute the genus as the sum of the genuses for the connected components.

The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. Following Stembridge [1, cor.4.7], the highest weight words indexing the irreducibles in $\mathfrak{gl_n}^{\otimes r}$ are ''staircase tableaux'' of length $2r$: sequences $(\gamma^{(0)},\dots,\gamma^{(2r)})$ of vectors in $\mathbb Z^n$ with decreasing entries, such that $\gamma^{(2i+1)}$ is obtained from $\gamma^{(2i)}$ by adding a unit vector and $\gamma^{(2i)}$ is obtained from $\gamma^{(2i-1)}$ by subtracting a unit vector. For $n=2$, the staircase tableaux whose final element is the zero vector are in natural correspondence with Dyck paths: adding the first or subtracting the second unit vector is translated to an up step, whereas adding the second or subtracting the first unit vector is translated to a down step. A Dyck path can be transformed into a ''bicoloured Motzkin path'' by replacing double up steps (double down, up-down, down-up steps) with up steps (down, coloured level, level steps). Note that the resulting path cannot have coloured level steps at height zero. In this context, say that a bicoloured Motzkin path has a $\mathfrak{gl}_2$-''descent'' between the following pairs of steps: * an up step followed by a level step * an up step followed by a down step, if the final height is not zero * a coloured level step followed by any non-coloured step. Then, conjecturally, the quasisymmetric expansion of the Frobenius character of the symmetric group $\mathfrak S_r$ acting on $\mathfrak{gl}_2^{\otimes r}$, is $$ \sum_M F_{Des(M)}, $$ where the sum is over all length $r$ prefixes of bicoloured Motzkin paths, $Des(M)$ is the set of indices of descents of the path $M$ and $F_D$ is Gessel's fundamental quasisymmetric function. The statistic recorded here is the number of $\mathfrak{gl}_2$-descents in the bicoloured Motzkin path corresponding to the Dyck path. Restricting to Motzkin paths without coloured steps one obtains the quasisymmetric expansion for the Frobenius character of $\mathfrak S_r$ acting on $\mathfrak{sl}_2^{\otimes r}$. In this case, the conjecture was shown by Braunsteiner [2].

The number of even non-empty partial sums of an integer partition.

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 number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.

The maximal cardinality of a set of vertices with the same neighbourhood in a graph. The set of so called mating graphs, for which this statistic equals $1$, is enumerated by [1].

The number of perfect matchings of a graph. A matching of a graph $G$ is a subset $F \subset E(G)$ such that no two edges in $F$ share a vertex in common. A perfect matching $F'$ is then a matching such that every vertex in $V(G)$ is incident with exactly one edge in $F'$.