The sorting index of a permutation. The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols. Compare this to [[St000018]], the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.

Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$. 2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.

The "bounce" of a permutation.

The Denert index of a permutation. It is defined as $$ \begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*} $$ where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].

The sum of the depths of the vertices (or total internal path length) of a binary tree. The depth of a vertex is the number of edges to the tree's root, see Section 2.3.4.5 of [1] and [3]. This statistic is the very first entry of the OEIS, see [2]. Observe that there the term '''height''' is used instead.

Babson and Steingrímsson's statistic of a permutation. In terms of generalized patterns this is $$ (13-2) + (21-3) + (32-1) + (21). $$ Here, $(\pi)$ denotes the number of times the pattern $\pi$ occurs in a permutation, and letters in the pattern which are not separated by a dash must appear consecutively.

The sum of the hook lengths in the first row of an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition. Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.

The aid statistic in the sense of Shareshian-Wachs. This is the number of admissible inversions [[St000866]] plus the number of descents [[St000021]]. This statistic was introduced by John Shareshian and Michelle L. Wachs in [1]. Theorem 4.1 states that the aid statistic together with the descent statistic is Euler-Mahonian.

Gives the vector space dimension of the homomorphism space between J^2 and J^2.