Jean Michel's site seems down, and while there is the archive.org version at web.archive.org/web/20240214180759/https://webusers.imj-prg.‌​fr/… , I'm wondering what is the actual home of the current version.

I don't know the history, but it's already called $f$ in Alfred Young's QSA (On Quantitative Substitutional Analysis) III paper (1928).

Made a few changes for clarity, hopefully getting your intent right. Nice answer!

@JoséFigueroa-O'Farrill: Link broken again; might be worth putting it on a safer place (e.g. zenodo).

And yeah, my mod-$p$ motivation seems to be unrelated to the actual question, and is quasi-answered in Künzer's thesis anyway (see my EDIT).

Very nice! I've just checked with SageMath that your permutation avoidance class has size $\dbinom{2n-2}{n-1}$ for all $n \leq 8$; this greatly improves the chances that you did things right. (Though some other classes have the same size, such as the slightly more memorable "avoid all $4$-patterns starting with a $4$ except for $4321$ and $4123$".) This actually makes me wonder if some Wilf equivalences can be explained by the invariance of the dimension applied to a quotient vector space of the symmetric group algebra, with two different confluent rewrite rules.

Yes. See, e.g., the proof of Lemma 6.5 in arxiv.org/abs/1509.08355v3 for a similiar situation (but without flagging).

(I have taken the liberty to replace inverse matrices by adjugate matrices here, though strictly speaking this slightly weakens the notion of diagonalizability. I think it makes for a nicer problem :)

The "cleanest" argument from the viewpoint of algebraic combinatorics would be showing that diagonalizable matrices are scheme-theoretically Zariski-dense in all matrices, i.e., that the algebra morphism from the coordinate ring of $R^{n\times n}$ (where $R$ is our base ring) to the coordinate ring of $R^{n\times n} \oplus R^n$ that evaluates each regular function (= polynomial in the entries of an $n\times n$-matrix) $f$ at the matrix product $A \operatorname{diag}\left(b_1,b_2,\ldots,b_n\right) \operatorname{adj}A$ is injective. Maybe there is a SAGBI basis for that?

By the way, are you sure about $s_{z/y}^{*}(c,d,-p)=\det((-1)^{z_{i}-y_{j}-i+j}e_{z_{i}-y_{j}-i+j} (p_{i+1},...,p_{N}) )$? This makes it sound like the entries in column $i$ should lie between $i+1$ and $N$, whereas your definition makes them lie between $1$ and $N-i$. Or does your $-p$ include flipping the order of the $p$'s as well?

The simplest approach appears to be considering the semistandard tableaux of shapes $\lambda / \nu$ and $\nu / \mu$ as one single semistandard tableau of shape $\lambda / \mu$, and fixing the latter. Then, the only sum that remains is a sum over all possible partitions $\nu$ that lie between $\mu$ and $\lambda$ and make the flagging conditions true for our tableau. Is this sum always $1$ or $0$ ? Can we prove it by toggling a single cell between the above-$\nu$ and below-$\nu$ parts? We would need to find a toggleable cell $\left(i,j\right)$ with entry lying in $\left[i+1,N-j\right]$.

Note that the de Longchamps point of a triangle is the reflection of its orthocenter in its circumcenter. Since the circumcenters of all four triangles are identical, this makes claim 2 a lot less surprising (the fact that the orthocenters form a quadrilateral anti-congruent to $ABCD$ is, I think, well-known). Claim 1 is the interesting one.

This looks a lot like a case for sign-reversing involutions.