The number of permutations with the same descent word as the given permutation. The descent word of a permutation is the binary word given by [[Mp00109]]. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map [[Mp00109]] containing a given permutation. This statistic appears as ''up-down analysis'' in statistical applications in genetics, see [1] and the references therein.

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 permutations whose descent word is the given binary word. This is the sizes of the preimages of the map [[Mp00109]].

The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. The total number of words with letter multiplicities given by an integer partition is [[St000048]]. For example, there are twelve words with letters $0,0,1,2$ corresponding to the partition $[2,1,1]$. Two of these contain the pattern $132$: $0,0,2,1$ and $0,2,1,0$. Note that this statistic is not constant on compositions having the same parts. The number of words of length $n$ with letters in an alphabet of size $k$ avoiding the consecutive pattern $132$ is determined in [1].

Number of parabolic noncrossing partitions indexed by the composition. Also the number of elements in the $\nu$-Tamari lattice with $\nu = \nu_\alpha = 1^{\alpha_1} 0^{\alpha_1} \cdots 1^{\alpha_k} 0^{\alpha_k}$, the bounce path indexed by the composition $\alpha$. These elements are Dyck paths weakly above the bounce path $\nu_\alpha$.

The number of standard Young tableaux of the skew partition.

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

The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. According to [1], this statistic was studied by Doron Gepner in the context of conformal field theory.

A descent variant minus the number of inversions. This statistic is defined for general finite crystallographic root system $\Phi$ with Weyl group $W$ as follows: Let $2\rho = \sum_{\beta \in \Phi^+} \beta = \sum_{\alpha\in\Delta}b_\alpha \alpha$ be the sum of the positive roots expressed in the simple roots. For $w \in W$ this statistic is then $$\operatorname{stat}(w) = \sum_{\alpha\in\Delta\,:\,w(\alpha) \in \Phi^-}b_\alpha - \ell(w)\,,$$ where the sum ranges over all descents of $w$ and $\ell(w)$ is the Coxeter length. It was shown in [1], that for irreducible groups, it holds that $$\sum_{w\in W} q^{\operatorname{stat}(w)} = f\prod_{\alpha \in \Delta} \frac{1-q^{b_\alpha}}{1-q^{e_\alpha}}\,,$$ where $\{ e_\alpha \mid \alpha \in \Delta\}$ are the exponents of the group and $f$ is its index of connection, i.e., the index of the root lattice inside the weight lattice. For a permutation $\sigma \in S_n$, this becomes $$\operatorname{stat}(\sigma) = \sum_{i \in \operatorname{Des}(\sigma)}i\cdot(n-i) - \operatorname{inv}(\sigma)\,.$$

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 $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.