The number of upper covers of a partition in dominance order.

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

The beta invariant of the graph. The beta invariant was introduced by Crapo [1] for matroids. For graphs with $n$ vertices the beta invariant is $$\beta(G) = (-1)^{n-c}\sum_{S\subseteq E} (-1)^{|S|} (n-c(S)),$$ where $c(S)$ is the number of connected components of the subgraph of $G$ with edge set $S$. For graphs with at least one edge the beta invariant equals the absolute value of the derivative of the chromatic polynomial at $1$. [2] The beta invariant also coincides with the coefficient of the monomial $x$, and also with the coefficient of the monomial $y$, of the Tutte polynomial.

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

The number of symmetric hooks on the diagonal of a partition.

The minimal number of edges to add or remove to make a graph edge transitive. A graph is edge transitive, if for any two edges, there is an automorphism that maps one edge to the other.

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.

The number of parts from which one can substract 2 and still get an integer partition.

The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.