The number of short braid edges in the graph of braid moves of a permutation. Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word. For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for $$(1243) = (12)(34)(23) = (34)(12)(23)$$ share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$. This statistic counts the number of such short braid moves among all reduced words.

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.