Computation and sampling for Schubert specializations.

At n = 17, Schubert principal specializations for RBPDs with a fixed boundary permutation are not maximized by layered permutations, disproving a Merzon–Smirnov conjecture.

• Why it matters: The conjecture’s predicted extremal structure links Schubert polynomial specializations to permutation classes, but counterexamples show the relationship is subtler than expected.
• What they did: The authors compute principal specializations using four recurrences (Macdonald, Lascoux–Schützenberger, Knutson, and Billey–Jockusch–Stanley) and run simulations and MCMC sampling of random permutations weighted by the specialization.
• The result: They still prove bounds supporting a refined asymptotic conjecture, observe a permuton-like limit in simulations, and develop an MCMC sampler (exact uniform sampling remains open) reaching n = 20 for computation and n = 100 for sampling.

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