The reversal length of a permutation. A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$. This statistic is then given by the minimal number of reversals needed to sort a permutation. The reversal distance between two permutations plays an important role in studying DNA structures.

The number of big ascents of a permutation after prepending zero. Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$. It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]). G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]]. [[St000646]] is the statistic without the border condition $\pi(0) = 0$.

The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.

The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].

The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$. The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$. One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$. Examples: * For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.

The number of elements generated by the Dyck path as a map in the full transformation monoid. We view the resolution quiver of a Dyck path (corresponding to an LNakayamaalgebra) as a transformation and associate to it the submonoid generated by this map in the full transformation monoid.

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

The number of left outer peaks of a permutation. A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$. In other words, it is a peak in the word $[0,w_1,..., w_n]$. This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].

The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. This descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].