Computation and sampling for Schubert specializations

Abstract: We present computational results on principal specializations $\mathfrak{S}w(1^n)$ of Schubert polynomials, which count reduced pipe dreams and reduced bumpless pipe dreams (RBPD). We find the first counterexample, at $n=17$, to the Merzon-Smirnov conjecture (arXiv:1410.6857) that the maximum of $\mathfrak{S}_w(1^n)$ over $S_n$ is attained at a layered permutation. The simulations suggest that $\lim{n \to \infty} \log(\max_{w\in S_n}\mathfrak{S}w(1^n))/n^2$ equals the maximal layered permutations' constant from Morales-Pak-Panova (arXiv:1805.04341). We also explore the random permutation drawn from the distribution proportional to $\mathfrak{S}_w(1^n)$, revealing permuton-like asymptotics similar to those for Grothendieck polynomials by Morales-Panova-Petrov-Yeliussizov (arXiv:2407.21653). We implement and compare three recurrences for $\mathfrak{S}_w(1^n)$: the descent formula (Macdonald), transition formula (Lascoux--Schutzenberger), and cotransition formula (Knutson). For sampling uniformly random RBPDs (whose count is $\sum{w\in S_n} \mathfrak{S}_w(1^n)$), we show that reducedness breaks the sublattice property of the ASM lattice, preventing monotone CFTP and causing false coalescence. We develop an efficient MCMC sampler with macroscopic "droop" updates for connectivity and fast mixing. Our code computes $\mathfrak{S}_w(1^n)$ up to $n\sim 20$ and samples random RBPDs up to $n\sim 60$ on a personal computer ($n\sim 100$ on a cluster).

• The paper develops three distinct recurrence relations and MCMC algorithms that efficiently compute principal specializations of Schubert polynomials.
• The paper experimentally disproves the Merzon–Smirnov conjecture by constructing explicit counterexamples that exceed the expected maximal values.
• The paper uncovers asymptotic phenomena, including limit shapes and permuton behavior, offering new insights into the combinatorial structure of reduced bumpless pipe dreams.

Detailed Summary and Analysis of "Computation and sampling for Schubert specializations" (2603.20104)

Introduction and Motivation

The study centers on the combinatorial and computational aspects of principal specializations of Schubert polynomials $\mathfrak{S}_w(1^n)$, a central object in algebraic combinatorics and geometry due to their role as a basis for the cohomology of flag varieties and their deep enumerative properties. Specifically, the focus is on the enumeration of reduced (bumpless) pipe dreams (BPDs, RBPDs) with prescribed permutation boundaries and on the typical structure of such sampled objects. The work tackles open conjectures on maximizing these specializations, investigates sampling and Markov chain dynamics on RBPDs, and provides efficient algorithms and empirical evidence for limit shapes and asymptotic phenomena.

Combinatorial Models: Pipe Dreams and Reduced Bumpless Pipe Dreams

Two principal combinatorial models are established for Schubert polynomial evaluations:

• Reduced Pipe Dreams (RC-graphs): Configurations of tiles in a staircase-shaped grid whose boundary permutation is traced via the path of colored pipes. Each pair of pipes can cross at most once—quantifying reducedness.
• Reduced Bumpless Pipe Dreams (RBPDs): Tilings of the $n\times n$ grid with six types of tiles, subject to the same reducedness condition. Without this constraint, the model coincides with the well-studied six-vertex model or alternating sign matrices (ASMs).

  Figure 1: The six tile types of bumpless pipe dreams (top) and an example of a reduced bumpless pipe dream of size $n=4$ with boundary permutation $w=2143$ (bottom).

Algorithmic Framework: Recurrences for $\mathfrak{S}_w(1^n)$

To compute principal specializations, the study implements and benchmarks three recurrrence relations:

• Descent Recurrence (Macdonald): Expresses $\Upsilon_w$ as a sum over descents with division by the length $\ell(w)$, requiring rational arithmetic. BFS level-order and memoization optimize computation.
• Transition Recurrence (Lascoux–Schützenberger): Relates $\Upsilon_w$ to those of permutations of lower length via $132$-pattern indices; allows fast DFS implementations without division.
• Cotransition Recurrence (Knutson): Based on Bruhat covers, this implements a level-order BFS relying solely on addition—well suited for exact integer arithmetic and large-scale parallelization.

Extensive performance comparisons are provided, demonstrating that the transition and cotransition formulas outperform the descent approach for generic or random $w$ (especially for typical RBPD-sampled permutations), though descent remains competitive for permutations with few inversions.

Disproof of the Merzon–Smirnov Conjecture

A highlight is the empirical disproof of the Merzon–Smirnov conjecture, which held that the maximal specialization $\max_{w\in S_n}\mathfrak{S}_w(1^n)$ always occurs at layered permutations—permutations composed of contiguous decreasing blocks. For $n\leq13$ this was exhaustively verified, but explicit counterexamples are constructed for $n=17$ and larger, with explicit permutations given that yield $\Upsilon_w$ exceeding the maximal value attained on layered ones by up to $15.6\%$. However, asymptotic evidence suggests that the exponential growth rate is unchanged, with the same constant as for layered permutations.

Sampling Algorithms and Markov Chain Methods

The structure and mixing properties of the state space of RBPDs are analyzed extensively:

• Obstacle to CFTP: The reducedness constraint invalidates the sublattice property within ASMs, leading to broken monotonicity and false coalescence in Coupling From The Past (CFTP) algorithms. Empirical results demonstrate significant bias if CFTP is (mis)applied to RBPDs (see Table 2 in the paper).
• MCMC Algorithms: The study develops a specialized Markov chain informed by local $2\times 2$ flips and non-local block moves (droops and undroops). Inclusion of droop moves is proven to ensure ergodicity and full connectivity, with droops corresponding to rectangular manipulations which overcome local energy "traps".

  Figure 2: Trajectories of the permutation length $\ell(w)$ during MCMC runs, indicating mixing behavior for various rectangle move distributions at $n=60$.

  

Figure 3: A stuck chain started from the identity RBPD at $n=60$; some locations remain frozen even after extensive mixing due to local traps.

Asymptotic and Limit Shape Phenomena

Using highly optimized MCMC sampling (up to $n=100$), the authors obtain strong empirical evidence for the existence and qualitative properties of the macroscopic limit shape (arctic curve) and conjectural Schubert permuton:

• Permuton Phenomenon: The rescaled empirical permutation matrix accumulates on a conical region with singular mass along its boundary, reminiscent of the arctic circle phenomena in integrable tiling models (cf. domino and lozenge tilings).

  Figure 4: The average permutation matrix $M$ at $n=100$, visually suggesting permuton convergence on a domain with singular arcs.

• Arctic Curve and Frozen Regions: The mixed derivative of the mean height function identifies the liquid/frozen interface, and matches the region where permutation matrix mass accumulates.

Figure 5: Mixed derivative $\Delta_x \Delta_y \bar{h}$ outlining the arctic curve of the liquid region, and its overlap with the nonzero values in $M$.

• Height Fluctuations: Analysis of sample-level fluctuations in the height function reveals freezing in the outer regions and potentially logarithmic variance scaling, suggestive of Gaussian free field fluctuations in the liquid zone.

Figure 6: Left, fluctuations in the height function for a random MCMC sample at $n=100$; right, the corresponding RBPD with complex pipe arrangements.

Theorems, Bounds, and Structural Observations

Bounds on Specializations with Local Moves

Explicit bounds are provided for the ratio $\Upsilon_{u}/\Upsilon_{w}$ when $u$ is obtained from $w$ by a single adjacent transposition, with lower and upper bounds depending linearly on the position. This implies that permutations close in Cayley distance exhibit similar asymptotic growth of $\mathfrak{S}_w(1^n)$, supporting the conjecture that layered permutations (although not always maximal) are sufficient for capturing the exponential rate of growth.

Consequences for Permutation Statistics

If $w$ is maximizer of $\Upsilon_w$, then necessarily $\mathrm{maj}(w)\geq \ell(w)$. However, the explicit maximizers found empirically appear to resist simple structural description (no block or avoidance pattern), revealing a more complex optimization landscape as $n$ grows.

Open Problems and Future Directions

Several open problems and avenues for further research are enumerated:

• Existence and Characterization of the Limit Schubert Permuton: The evidence supports convergence to a permuton with a singular boundary; precise analytic characterization is open.
• Explicit Arctic Curve Equation: The empirical boundary separating the frozen and liquid zones in configurations lacks an analytic description unlike the integrable $\beta=1$ (Grothendieck) case.
• Mixing Times and Algorithmic Complexity: While MCMC shows empirical polynomial mixing in some regimes, rigorous upper/lower bounds and dependence on move set remain unproven.
• Generalizations to $q$-specializations and Other Deformations: Extensions to $q$-weighted models and double Schubert polynomials, as well as other measures such as symmetric products, are mentioned as fertile ground.
• Exact Sampling in Non-integrable Regimes: While CFTP is provably infeasible for RBPDs under current constraints, new combinatorial approaches or non-local shuffling algorithms could yield unbiased sampling.

Implications and Prospective Developments

Theoretical Implications

The study advances the understanding of the asymptotics, typical structure, and optimization problems for Schubert polynomial specializations, bridging enumerative combinatorics and statistical mechanics. The proof techniques and algorithmic reductions connect closely to recent advances in integrable probability, but the non-local reducedness constraint places RBPDs outside the currently tractable scope of integrable systems, inviting the development of new probabilistic and combinatorial methods.

Practical Algorithmics

The implementation of efficient, large-scale, and robust recurrence relations and MCMC sampling tools makes previously infeasible computational experiments accessible, with code released publicly for further exploration and validation. This opens the door to systematic statistical investigations and may catalyze further discoveries in related high-dimensional models beyond symmetric group enumeration.

Directions for AI and Machine-Assisted Enumeration

The authors note the value of contemporary AI tools for code optimization and hypothesis generation (e.g., the use of ChatGPT and Claude Code), as well as recent use of large ML models (FunSearch) for heuristic counterexample-finding. As computational and AI-assisted mathematics matures, the interface with combinatorial and asymptotic problems of this nature will become increasingly prominent.

Conclusion

This work elucidates the computation and sampling of Schubert principal specializations through a blend of combinatorial, algorithmic, and probabilistic perspectives. It refutes a leading structural conjecture, advances the state-of-the-art for both computation and sampling of RBPDs, and presents substantial empirical evidence regarding limit shapes and fluctuations. The identification of structural obstacles to exact sampling in non-integrable models, as well as the asymptotic stability of layered-structure maxima, pave the way for further exploration in enumerative geometry, statistical physics-style random tiling models, and the development of novel, large-scale combinatorial enumeration methods.