EDIT: We now know how to prove the following and will update the answer with an outline.
FindStat's original answer
Then it appears that the number of $2$-Gorenstein failures is the composition of http://findstat.org/Mp00129 (The Billey-JokuschJockusch-Stanley bijection to 321-avoiding permutations) and http://findstat.org/St000732 (The number of double deficiencies of a permutation).
Outline of proof
HereThroughout the proof, a Dyck path starts at the origin, has north and east steps and never goes below the diagonal $y=x$. We think of the Dyck path as running in an array of columns labelled $1$ to $n$ from left to right, and rows labelled $1$ to $n$ from bottom to top.
Let us first recall the bijection due to Billey, Jockusch and Stanley, sending a Dyck path $D$ to a $321$-avoiding permutation $\pi$. To obtain the (horizontally flipped) permutation matrix of $\pi$ corresponding to $D$, put a cross into each valley (an east step followed by a north step) and fill the remaining slots with an increasing sequence of crosses.
Our claim is an immediate consequence of the Sage code usingfollowing three statements.
Lemma 1. $c_{x+1} = c_x - 1$ if and only if $x$ is a weak deficiency of $\pi$, that is $\pi(x)\leq x$. Equivalently, there is a double east step spanning columns $x$ and $x+1$.
Lemma 2. $c_{x+1} = c_x - 1$ and $c_{x+1} + d_x = d_{x+c_{x+1}}$ if and only if $x$ is a fixed point of $\pi$.
Lemma 3. Let $x$ be a strict deficiency of $\pi$. Then $\pi^{-1}(x)$ is also a strict deficiency if and only if $x+1=a+b$ where $a$ is a descent and $b\in X_a$.
We intend to provide a detailed and illustrated version of this argument within the FindStat interface:next few days.
Sage code used to discover the conjecture
