The record statistic and forward stability of Schubert products

Abstract: We initiate a probabilistic study of forward stability for products of Schubert polynomials through the record statistic (left-to-right maxima) of permutations. Building on the explicit record formula for forward stability obtained by Hardt and Wallach, we study random pairs of permutations drawn from three natural families: uniform permutations, Grassmannian permutations, and Boolean permutations. For each family, we determine record probabilities and use them to analyze the asymptotic behavior of forward stability. For uniform and Grassmannian permutations, we obtain asymptotics for the mean together with limiting distribution results. For Boolean permutations, we prove linear-order growth of the mean, and our analysis also produces an explicit time-inhomogeneous Markov chain that yields an exact linear-time uniform sampler. Beyond these cases, we prove that the record-set statistic is equidistributed on the avoidance classes of $132$ and $231$, and consequently the corresponding forward stability distributions coincide. We conclude with conjectures for numerous further permutation classes and a conjectural recursive criterion for when two avoidance classes have the same record-set distribution.

• The paper introduces a record-based max formula linking left-to-right record statistics with the forward stability of Schubert products in cohomology.
• It analyzes uniform, Grassmannian, and Boolean permutation families, providing explicit formulas and asymptotic distributions for record occurrences.
• The study bridges probabilistic combinatorics and algebraic geometry, yielding new insights into Schubert calculus and pattern-avoidance equidistribution.

The Record Statistic and Forward Stability of Schubert Products

Introduction and Motivation

This paper investigates the intersection of probabilistic combinatorics and algebraic geometry by initiating a systematic analysis of the record statistic (left-to-right maxima) within permutation families, and its role in determining the forward stability of products of Schubert polynomials. It leverages a max formula for the forward stability function $\mathrm{FS}(u, v)$—the minimal rank $N$ for which the Schubert product $\mathfrak{S}_u \mathfrak{S}_v$ is fully realized in the cohomology ring of the flag variety $GL_N/B$—to recast a geometric stabilization question in terms of extremal statistics for records in random permutations.

The primary focus is the asymptotic and distributional behavior of $\mathrm{FS}(u, v)$ for independent random permutations $(u, v)$ from three fundamental families: uniform permutations, Grassmannian permutations (at most one descent), and Boolean permutations (no repeated generators in any reduced word). The analysis is extended to questions of record-set equidistribution, particularly for pattern-avoidance classes, which reveals regimes of universal stabilization behavior linked to record density.

Record Statistic: Framework and Explicit Formulas

For $w \in S_n$, a record at position $j$ is defined as the absence of a left inversion: $d_j(w) = 0$ where $d_j(w)$ is the count of entries left of $N$0 that are greater than $N$1. The indicator $N$2 records this event, and its complement $N$3 is $N$4 if $N$5 is not a record.

The paper provides explicit closed formulas for the record probabilities across the three permutation families:

• Uniform: $N$6 (R\'enyi's theorem).
• Grassmannian:

$N$7

• Boolean: An explicit formula involving Fibonacci numbers:

$N$8

where $N$9 is the $\mathfrak{S}_u \mathfrak{S}_v$0th Fibonacci number.

For Boolean permutations, the enumeration of record occurrences relies on structural decomposition into blocks in the reduced word, leading to a recursion resolved via Fibonacci identities.

Forward Stability via Record Max Formula

By leveraging results of Hardt and Wallach, the forward stability is determined by a record-based process:

$\mathfrak{S}_u \mathfrak{S}_v$1

where $\mathfrak{S}_u \mathfrak{S}_v$2 counts the non-record positions $\mathfrak{S}_u \mathfrak{S}_v$3 in $\mathfrak{S}_u \mathfrak{S}_v$4. Thus, forward stability is reduced to the maximization of a random walk built from sums of local record indicators, converting the geometric stabilization problem for Schubert products into an instance of extreme value theory for dependent combinatorial processes.

Asymptotics and Fluctuations in Major Families

Uniform Permutations

For $\mathfrak{S}_u \mathfrak{S}_v$5,

• Mean: $\mathfrak{S}_u \mathfrak{S}_v$6, with $\mathfrak{S}_u \mathfrak{S}_v$7 and $\mathfrak{S}_u \mathfrak{S}_v$8 the Euler–Mascheroni constant.
• Distribution:

$\mathfrak{S}_u \mathfrak{S}_v$9

This demonstrates that, although records are rare, the large deviations of their sum still obey a central limit theorem, and the stabilization window grows nearly linearly, with a logarithmic correction.

Grassmannian Permutations

For $GL_N/B$0,

• Mean: $GL_N/B$1.
• Distribution:

$GL_N/B$2

with $GL_N/B$3 i.i.d. standard normal.

Here, compared to the uniform case, the expected stabilization is less than $GL_N/B$4, reflecting the higher density of records in Grassmannian permutations.

Boolean Permutations

For $GL_N/B$5,

• Mean: $GL_N/B$6.
• Structural insight: The underlying stochastic process admits a finite-state inhomogeneous Markov encoding, leading to an explicit, linear-time uniform sampler, and the forward stability statistic is maximally concentrated close to $GL_N/B$7.

This is a striking contrast: in a regime of very dense records (almost every position), the stabilization is almost minimal.

Backward Stability and Distributional Equivalence

Backward stability $GL_N/B$8 is shown to satisfy

$GL_N/B$9

where $\mathrm{FS}(u, v)$0 is the reverse permutation. Since $\mathrm{FS}(u, v)$1 preserves permutation class for all main families considered, the distributional results for forward stability extend directly to backward stability after a deterministic shift.

Record Equidistribution and Pattern Classes

The analysis is extended to pattern-avoidance classes, categorizing asymptotic $\mathrm{FS}(u, v)$2 into three regimes—record-dense ($\mathrm{FS}(u, v)$3), intermediate ($\mathrm{FS}(u, v)$4 with $\mathrm{FS}(u, v)$5), and record-sparse ($\mathrm{FS}(u, v)$6)—across classical classes such as $\mathrm{FS}(u, v)$7-avoiding, $\mathrm{FS}(u, v)$8-avoiding, Boolean, and vexillary permutations. Monte Carlo estimations and combinatorial reasoning conjecture exact or nearly exact constants in these cases.

A remarkable theoretical result is established: the record-set distributions on $\mathrm{FS}(u, v)$9 and $(u, v)$0 are identical for all $(u, v)$1, leading to the exact matching of the forward stability distributions for random pairs drawn from either class. This is proved via a recursive decomposition giving a product formula in terms of Catalan numbers for the fiber cardinality of record sets.

The paper formulates a recursive criterion conjecture: two patterns define avoidance classes with the same record-set distribution if and only if their minors (by deletion neither at the maximum nor minimal positions) are pairwise record-equivalent, along with Wilf-equivalence and initial record set matching.

Probabilistic and Combinatorial Techniques

The proofs integrate probabilistic tools adapted to non-independent, non-stationary processes, such as geometric strong mixing, tailored versions of the Bernstein inequality for block sums (using Merlevède–Peligrad–Rio techniques), and explicit combinatorial enumeration via block decompositions, tag words, and Fibonacci or Catalan number recurrences. For Boolean permutations, an explicit Markov-chain uniform sampler is constructed, highlighting the role of local-to-global combinatorics in the permutation’s record structure.

Implications, Theoretical Insights, and Future Directions

The explicit connection between record statistics—a classical area in random permutations—and stabilization problems in algebraic geometry broadens the toolbox for studying polynomial stability and structure constants in Schubert calculus. The probabilistic framework enables sharp determination of stabilization thresholds, reveals universality classes of permutation families under Schubert product stabilization, and yields new exact combinatorial identities (e.g., equidistribution between $(u, v)$2 and $(u, v)$3).

Potential extensions include:

• Proving the conjectured classifications for broader pattern-avoidance classes and connecting stabilization behavior to representation-theoretic or geometric invariants.
• Generalizing the probabilistic analysis to more refined structure constants in cohomology, Grothendieck, or $(u, v)$4-theoretic settings, where initial results are mentioned.
• Applying these methods to random walks and extreme value theory for general dependent combinatorial processes beyond records.

Conclusion

This paper establishes a rigorous probabilistic-combinatorial paradigm for quantifying Schubert product stabilization, driven by the distributional behavior of left-to-right maxima in various permutation classes. It yields explicit asymptotic and distributional laws, precise combinatorial identities, and an avenue for further investigations into the interplay of symmetric group statistics, pattern avoidance, and algebraic geometry of flag varieties (2604.02964).