The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.

The cycle descent number of a permutation. Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.

The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.

The number of big exceedences of a permutation. A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$. This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.

The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path.

The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.

The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.

The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$ such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$. Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation. An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows. Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.

The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].

The number of bonds in a permutation. For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.