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OP-9571Status unassessedCombinatorics

Tableaux problem 11

Tableaux problem 11

MSC 05 · Combinatorics

Inferred from the community field; finer classification has not been reviewed.

Specialist source · status unassessedAs of 2026-09-10Source & evidence ↓
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Problem statement

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

14≤lim inf⁡n→∞log⁡2u(n)n2≤lim sup⁡n→∞log⁡2u(n)n2≤12.\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≤α≤1/21/4\leq\alpha\leq 1/2 such that u(n)∼2αn2u(n)\sim 2^{\alpha n^{2}}?

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

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Background

Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.

Dale R. Worley, version 3 (5 April 2026). Source citation keys are retained even where the arXiv HTML does not resolve its bibliography.

CC BY 4.0; Dale R. Worley. HTML converted to Markdown with attribution and source links.

References
  1. [1]Tableaux — A notebook of open problems — Tableaux problem 11 link
  2. [2]Creative Commons Attribution 4.0 International link

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