The sum of the hook lengths 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 + 1. This statistic is the sum of all hook lengths of a partition.

The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.

The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra.

The cardinality of the Frattini sublattice of a lattice. The Frattini sublattice is the intersection of all proper maximal sublattices of the lattice.

The number of doubly irreducible elements of a lattice. An element $d$ of a lattice $L$ is '''doubly irreducible''' if it is both join and meet irreducible. That means, $d$ is neither the least nor the greatest element of $L$ and if $d=x\vee y$ or $d=x\wedge y$, then $d\in\{x,y\}$ for all $x,y\in L$. In a finite lattice, the doubly irreducible elements are those which cover and are covered by a unique element.

The comajor index for set-valued two-row standard Young tableaux. The comajorindex is the sum $\sum_k (n+1-k)$ over all natural descents $k$. Bijections via bicolored Motzkin paths (with two restrictions, see [1]) give the following for Dyck paths. Let $j$ be smallest integer such that $2j$ is a down step. Then $k$ is a natural descent if * $k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a valley i.e. [0,1], or * $k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a peak i.e. [1,0], or * $k-1\ge j$ and positions $2(k-1),2k-1,2k$ form [0,1,1], or * $k=j$ and positions $2k-1,2k$ are double down i.e. [0,0], or * $k < j$ and positions $2k-1,2k$ are a valley i.e. [0,1].

The sum of the entries of the Gelfand-Tsetlin pattern.

The sum of the entries of a parking function. The generating function for parking functions by sum is the evaluation at $x=1$ and $y=1/q$ of the Tutte polynomial of the complete graph, multiplied by $q^\binom{n}{2}$.

The inversion sum of a binary word. A pair $a < b$ is an inversion of a binary word $w = w_1 \cdots w_n$ if $w_a = 1 > 0 = w_b$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.

The sum of the positions of the left to right maxima of a permutation. The generating function for this statistic is $$\sum_{\pi\in\mathfrak S_n} q^{slrmax(pi)} = \prod_{k=1}^n (q^k+k-1),$$ see [prop. 2.6., 1].