Layered permutations determine the exponential growth rate of max principal specializations

Prove that the exponential growth rate of max_{w∈S_n} 𝛶_w equals the growth rate of the maximum over layered permutations, i.e., lim_{n→∞} (1/n^2) log_2 max_{w∈S_n} 𝛶_w = lim_{n→∞} (1/n^2) log_2 max_{w layered} 𝛶_w ≈ 0.29.

Background

Although the Merzon–Smirnov conjecture (that layered permutations maximize 𝛶_w for each n) is disproved here starting at n=17, data and bounds suggest that the leading n2-scale exponential rate is unchanged by allowing all permutations.

The layered case admits explicit product formulas and optimization, yielding a known limiting constant; the conjecture asserts that any advantage of non-layered maximizers is subexponential in n2.

References

Conjecture The maximum of $\Upsilon_w$ over layered permutations has the same exponential growth rate as the maximum over the whole $S_n$, that is, $$\lim_{n\to \infty} \frac{1}{n2} \log_2 \,\max_{w \in S_n} \Upsilon_w \,=\, \lim_{n\to \infty} \frac{1}{n2} \log_2 \,\max_{w \text{ layered} \Upsilon_w \,\approx\, 0.29, $$ where the layered limit was computed in . In other words, any potential improvement over layered permutations is subexponential in $n2$.

— Computation and sampling for Schubert specializations  (2603.20104 - Anderson et al., 20 Mar 2026) in Section 6.6