The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is $$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$ for some $(r,a,b)$. This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].

The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.

The connected partition number of a graph. This is the maximal number of blocks of a set partition $P$ of the set of vertices of a graph such that contracting each block of $P$ to a single vertex yields a clique. Also called the pseudoachromatic number of a graph. This is the largest $n$ such that there exists a (not necessarily proper) $n$-coloring of the graph so that every two distinct colors are adjacent.

The number of minimal elements in Bruhat order not less than the signed permutation. The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent. This is the size of the essential set of the signed permutation, see [1].

The mutual visibility number of a graph. This is the largest cardinality of a subset $P$ of vertices of a graph $G$, such that for each pair of vertices in $P$ there is a shortest path in $G$ which contains no other point in $P$. In particular, the mutual visibility number of the disjoint union of two graphs is the maximum of their mutual visibility numbers.

The tree-depth of a graph. The tree-depth $\operatorname{td}(G)$ of a graph $G$ whose connected components are $G_1,\ldots,G_p$ is recursively defined as $$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$$ Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once). Tree-depth is strictly greater than [[St000536|pathwidth]]. A [[St001120|longest path]] in $G$ has at least $\operatorname{td}(G)$ vertices [3].

The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].

The bandwidth of a graph. The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$. We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.

The number of repeated entries in the Lehmer code of a permutation. The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.