The length of a longest palindromic factor of a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the maximal length of a palindromic factor.

The length of a longest palindromic subword of a binary word. A subword of a word is a word obtained by deleting letters. This statistic records the maximal length of a palindromic subword. Any binary word of length $n$ contains a palindromic subword of length at least $n/2$, obtained by removing all occurrences of the letter which occurs less often. This bound is obtained by the word beginning with $n/2$ zeros and ending with $n/2$ ones.

The minimal period of a binary word. This is the smallest natural number $p$ such that $w_i=w_{i+p}$ for all $i\in\{1,\dots,|w|-p\}$.

The minimal number such that all substrings of this length are unique.

The length of the longest alternating subword. This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.

The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.

The number of edges that can be added without increasing the maximal degree of a graph. This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on $n$ vertices, which has maximal degree $n-1$ and therefore has statistic $\binom{n-1}{2}$.

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 sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.

The column of the unique '1' in the last row of the alternating sign matrix.