The length of the longest weakly inreasing subsequence of parts of an integer composition.

The number of weak descents in an integer composition. A weak descent of an integer composition $\alpha=(a_1, \dots, a_n)$ is an index $1\leq i < n$ such that $a_i \geq a_{i+1}$.

The maximal multiplicity of an eigenvalue in a graph.

The number of stack-sorts needed to sort a permutation. A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series. Let $W_t(n,k)$ be the number of permutations of size $n$ with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).

The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b

The maximal number of nonzero entries on a diagonal of a permutation matrix. For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is $$\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix},$$ and the entries corresponding to $\pi_2=1$, $\pi_3=2$ and $\pi_5=4$ are all on the fourth diagonal from the right. In other words, this is $\max_k \lvert\{i: \pi_i-i = k\}\rvert$

The maximal sum of entries on a diagonal of an alternating sign matrix. For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right)$$ are $(0,1,1,0,1,1,0)$, so the statistic is $1$. This is a natural extension of [[St000887]] to alternating sign matrices.

The minimal number such that all substrings of this length are unique.

Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

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 to which we can associate a Dyck path as the top boundary of the Auslander-Reiten quiver of the LNakayama algebra. The statistic gives half the dominant dimension of hte first indecomposable projective module in the special CNakayama algebra.