Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. Associate to this special CNakayama algebra a Dyck path as follows: In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra. The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.

The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.

The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.

The index of the last row whose first entry is the row number in a standard Young tableau.

The first ascent of a permutation. For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$. For the first descent, see [[St000654]].

The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.

The number of leading ones in a binary word.

The Elizalde-Pak rank of a permutation. This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$. According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.

The number of runs of ones in a binary word.

The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.