The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].

The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.

The number of global descents of a permutation. The global descents are the integers in the set $$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$ In particular, if $i\in C(\pi)$ then $i$ is a descent. For the number of global ascents, see [[St000234]].

The number of occurrences of the vincular pattern |1-23 in a permutation. This is the number of occurrences of the pattern $123$, where the first two matched entries are the first two entries of the permutation. In other words, this statistic is zero, if the first entry of the permutation is larger than the second, and it is the number of entries larger than the second entry otherwise.

The exponent of a binary word. This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called '''frequency'''.

The length of the Lyndon factorization of the binary word. The Lyndon factorization of a finite word w is its unique factorization as a non-increasing product of Lyndon words, i.e., $w = l_1\dots l_n$ where each $l_i$ is a Lyndon word and $l_1 \geq\dots\geq l_n$.

The flex of a binary word. This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].

The number of borders of a binary word. A border of a binary word $w$ is a word which is both a prefix and a suffix of $w$.

The length of the border of a binary word. The border of a word is the longest word which is both a proper prefix and a proper suffix, including a possible empty border.

The decomposition (or block) number of a permutation. For $\pi \in \mathcal{S}_n$, this is given by $$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].