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on Tableaux problem 11

The statement

11.[Stan2017a]*sec. 1 via [Set] Define $\nu_{w}=\mathfrak{S}_{w}(1,\ldots,1)$ which is the sum of the coefficients of the Schubert polynomial $\mathfrak{S}_{w}$ of the permutation $w$ . How large can $\nu_{n}$ be for $w\in S_{n}$ ? Define $u(n)$ to be this maximum. We have some rather crude bounds which show that,

\frac{1}{4}\leq\liminf_{n\rightarrow\infty}\frac{\log_{2}u(n)}{n^{2}}\leq\limsup_{n\rightarrow\infty}\frac{\log_{2}u(n)}{n^{2}}\leq\frac{1}{2}.

Determine better bounds. That is, can we find $1/4\leq\alpha\leq 1/2$ such that $u(n)\sim 2^{\alpha n^{2}}$ ?

[MoraPakPan2019a]*Th. 1.3 improves the lower bound to $\approx 0.2932362762$ . [MoraPanPet2025a]*Rem. 6.8 improves the upper bound to 0.37.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.