The size of the overlap set of a permutation. For a permutation $\pi\in\mathfrak S_n$ this is the number of indices $i < n$ such that the standardisation of $\pi_1\dots\pi_{n-i}$ equals the standardisation of $\pi_{i+1}\dots\pi_n$. In particular, for $n > 1$, the statistic is at least one, because the standardisations of $\pi_1$ and $\pi_n$ are both $1$. For example, for $\pi=2143$, the standardisations of $21$ and $43$ are equal, and so are the standardisations of $2$ and $3$. Thus, the statistic on $\pi$ is $2$.

The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.

The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. Also the number of simple modules that are isolated vertices in the sense of 4.5. (4) in the reference.

The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module. For the first 196 values, the statistic also gives the number of indecomposable non-projective modules $X$ such that $\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.

Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module.

The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. For the projective dimension see [[St001007]] and for 1-regularity see [[St001126]]. After applying the inverse zeta map [[Mp00032]], this statistic matches the number of rises of length at least 2 [[St000659]].

The holeyness of a permutation. For $S\subset [n]:=\{1,2,\dots,n\}$ let $\delta(S)$ be the number of elements $m\in S$ such that $m+1\notin S$. For a permutation $\pi$ of $[n]$ the holeyness of $\pi$ is $$\max_{S\subset [n]} (\delta(\pi(S))-\delta(S)).$$