The number of ribbon shaped standard tableaux. A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape. This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed. For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals $$\sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell},$$ where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]

The number of standard Young tableaux of the skew partition.

The number of reduced words for a permutation. This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.

The number of reduced Kogan faces with the permutation as type. This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.

The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.

The number of connected components of long 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 long braid move if they are obtained from each other by a modification of the form $s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word. For example, the two reduced words $s_1s_3s_2s_3$ and $s_1s_2s_3s_2$ for $$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$$ share an edge because they are obtained from each other by interchanging $s_3s_2s_3 \leftrightarrow s_3s_2s_3$. This statistic counts the number connected components of such long braid moves among all reduced words.

The number of subgraphs. Given a graph $G$, this is the number of graphs $H$ such that $H \hookrightarrow G$.

The Hosoya index of a graph. This is the total number of matchings in the graph.

The number of occurrences of the pattern 123 or of the pattern 231 in a permutation.

The number of occurrences of the pattern 123 or of the pattern 312 in a permutation.