The maximal number of elements covered by an element in a poset.

The maximal number of elements covering an element of a poset.

The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

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 length of the longest constant subword.

The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

The number of big ascents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$. For the number of small ascents, see [[St000441]].

The number of cyclic descents of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.

The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,...,n). Let $\rho=(1,\dots,n)$ and $\sigma=(1,2)$. Then, for a permutation $\pi\in\mathfrak S_n$, this statistic is $$\min\{k\mid \pi=\rho^{i_0}\sigma\rho^{i_1}\sigma\dots\rho^{i_{k-1}}\sigma\rho^{i_k}, 0\leq i_0,\dots,i_k < n\}.$$ Put differently, it is the minimal length of a factorization into cyclic shifts of the transposition $(1,2)$ (see [[St001076]]) of any of the permutations $\rho^k\pi$ for $0\leq k < n$.