Computation and sampling for Schubert specializations
- 1Department of Mathematics, Ohio State University, Columbus, OH 43210
- 2Department of Mathematics, University of Southern California, Los Angeles, CA 90089
- 3Department of Mathematics, University of Virginia, Charlottesville, VA 22904
Abstract
We study a problem at the interface of enumerative algebraic geometry, algebraic combinatorics, and the statistical mechanics of vertex models with long-range interactions. We develop methods to explore the values Sw(1,…,1) of Schubert polynomials (called principal specializations), which count reduced bumpless pipe dreams (RBPDs) with boundary permutation w∈Sn. We exhibit counterexamples to the conjecture of Merzon and Smirnov that Sw(1n) is maximized on layered permutations; the smallest such example we find is at n = 17. Despite the counterexamples, we present bounds that support a refined conjecture that the maximum of Sw(1n) over all w∈Sn grows at the same asymptotic rate as the maximum over layered permutations (found by Morales, Pak, and Panova). We implement and benchmark four recurrences for Schubert specializations: the descent formula of Macdonald, the transition formula of Lascoux–Schützenberger, the cotransition formula of Knutson, and the formula of Billey, Jockusch, and Stanley. We also explore typical random permutations drawn with probability proportional to Sw(1n), equivalently from uniformly random RBPDs. Our simulations reveal a permuton-like limit analogous to the Grothendieck setting studied by Morales, Panova, Petrov, and Yeliussizov. For sampling, we show that the standard monotone Coupling From The Past framework does not apply to RBPDs, leaving exact uniform sampling open. To bypass this, we develop an efficient Markov chain Monte Carlo sampler augmented with macroscopic “droop” updates that preserve connectivity and accelerate mixing. Our publicly available fast implementations reach n = 20 for exact computation of Sw(1n) and n = 100 for Markov chain Monte Carlo sampling using a chain whose stationary distribution is uniform on RBPDs (run on a cluster).
Related
- Effective computation of Moore-Penrose inverses over fields of rational functions by specializations
- Euler obstructions and Verdier specializations
- Hilbertian Fields and Profinite Iterated Monodromy Groups Over Transcendental Elements with Specializations of Infinite Index
- Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density
- Principal specializations of the t-chromatic symmetric function
BibTeX
@article{anderson2026computation,
title = {{Computation and sampling for Schubert specializations}},
author = {David Anderson and Greta Panova and Leonid Petrov},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
year = {2026},
month = oct,
doi = {10.1073/pnas.2615757123},
url = {https://doi.org/10.1073/pnas.2615757123},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - JOUR
TI - Computation and sampling for Schubert specializations
AU - David Anderson
AU - Greta Panova
AU - Leonid Petrov
PY - 2026
DA - 2026/10/29
JO - Proceedings of the National Academy of Sciences of the United States of America
DO - 10.1073/pnas.2615757123
UR - https://doi.org/10.1073/pnas.2615757123
AB - We study a problem at the interface of enumerative algebraic geometry, algebraic combinatorics, and the statistical mechanics of vertex models with long-range interactions. We develop methods to explore the values Sw(1,…,1) of Schubert polynomials (called principal specializations), which count reduced bumpless pipe dreams (RBPDs) with boundary permutation w∈Sn. We exhibit counterexamples to the conjecture of Merzon and Smirnov that Sw(1n) is maximized on layered permutations; the smallest such example we find is at n = 17. Despite the counterexamples, we present bounds that support a refined conjecture that the maximum of Sw(1n) over all w∈Sn grows at the same asymptotic rate as the maximum over layered permutations (found by Morales, Pak, and Panova). We implement and benchmark four recurrences for Schubert specializations: the descent formula of Macdonald, the transition formula of Lascoux–Schützenberger, the cotransition formula of Knutson, and the formula of Billey, Jockusch, and Stanley. We also explore typical random permutations drawn with probability proportional to Sw(1n), equivalently from uniformly random RBPDs. Our simulations reveal a permuton-like limit analogous to the Grothendieck setting studied by Morales, Panova, Petrov, and Yeliussizov. For sampling, we show that the standard monotone Coupling From The Past framework does not apply to RBPDs, leaving exact uniform sampling open. To bypass this, we develop an efficient Markov chain Monte Carlo sampler augmented with macroscopic “droop” updates that preserve connectivity and accelerate mixing. Our publicly available fast implementations reach n = 20 for exact computation of Sw(1n) and n = 100 for Markov chain Monte Carlo sampling using a chain whose stationary distribution is uniform on RBPDs (run on a cluster).
SN - 0027-8424
SN - 1091-6490
ER -