Wow! I didn't think there was so deep math behind this question! Thanks for the elegant answer.

If $n$ is not prime, all the divisors $d$ appears in the product in the factor $(d+1)^2-1^2$. If $p$ is prime it is not true: for $p=2$, $V(1,4) = 4-1 = 3$ is odd. PS i am assuming @MaxAlekseyev is true, that is the only relevant permutation is $\pi_k =k$.

By using Dugger's cobar construction in $Top^{op}$ plus the following formula for homotopy totalization: ncatlab.org/nlab/show/homotopy+totalization one can indeed find the above formula, but I have not done all the calculations precisely. Since it is not straightforward and I'd like not to repeat this proof, I would still be interested in a reference. But many thanks for pointing to the right thing!!!

That's right. I think the concept I was missing is the one of weighted (co)-limit, which definitely depicts the kind of operation I was doing. Thanks!

Hi Fosco! Thank you very much. Could you write down more precisely the proof for $F$ mute? Since this is actually the case I have, I would like to include your proof in my article giving you due credit :)) I am not sure this is somewhere in the literature. I could just make the explicit proofs with my geometric objects (mines are simplicial sets) but I find it to be very inelegant if an abstract proof is available.

not yet, that's a good tip

For anyone who is interested, the answer should be yes at the end. I explicitly constructed the spectral sequence from the $d_i$'s only (which is coherent with the classical spectral sequence for strict cosimplicial objects). I still have to verify that is the same fo the lurie spectral sequence, though I would be very surprised that one is able to construct two substantially different spectral sequences in so much generality, that agree for trivial higher homotopies.

Very beatiful!!

Regarding the ItaCa group, thank you! I will join. Being honest, I was a fervid category theorist since the beginning of my university; then last year of my master I have studied infty-categories, naively thinking they were categories but more powerful. Nobody told me it was topology. So in the end I happened to become an algebraic topologist (which i like i lot though) with a categorical flavour, but I am more into geometry/homotopical algebra. Not sure I am well-suited for ItaCa but always up for nice catego-conversations :))

Thank you!! I am honored you are answering me :)) Yes, I agree with the weird-ness of using more variables, and I just thought about it a few hours after i posted. I like very much your reformulation with $+,-$ and it is coherent with my case (in which actually the initial bifunctor only depended on the contravariant component). And yes, being surjective and with set-fibers exactly means being a discrete cartesian fibration - that's how it popped up in my case. I wrote this observation on the homotopy theory chat, I will edit it in my question.

The link about 0,1 matrices is not that relevant to me, while the OEIS link for $2\times2$ matrices is nice. With a simple python program it's easy to see that the probability $p(n)$ of being non-singular with nonnegative coefficients up to $n$ is such that $p(n)/(1-1/n)$ is strictly decreasing and greater than $1$ (from some point on), so that one can conjecture that $p(n) \le (1+C)(1-1/n)$ for some positive constant $C<1/40$ (since for $n=31$ we get less than $1/40$. I'll look at the other links soon

Regarding the size of $m,n$: I'd like to have an estimate in both variables (like a taylor expansion in $1/m, 1/n$), but it's ok if we fix the size of the matrix and we let the bound for the coefficients go to infinity.

Yes, the question is somewhat vague... It was meant to be lighthearted :) I don't need this for any application, so a variant of the estimate is ok too!

@NoahSnyder: I think this easy proof for uniqueness pairs well with the Galois proof I gave at the end, giving an "easy" existence proof with no verifications needed.