The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.

The number of mid points of increasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$. The generating function is given by [1].

The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

The cardinality of the support of a permutation. A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$. The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$. Thus, the connectivity set is the complement of the support.

The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).$$

The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.

The prefix exchange distance of a permutation. This is the number of star transpositions needed to write a permutation. In symbols, for a permutation $\pi\in\mathfrak S_n$ this is $$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n\},$$ where $\tau_a = (1,a)$ for $2 \leq a \leq n$. [1, Lem. 2.1] shows that the this length is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$. One may find in [2] explicit formulas for its generating function and a combinatorial proof that it is asymptotically normal.

The smallest label of a leaf of the increasing binary tree associated to a permutation.