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arxiv: 1907.11802 · v1 · pith:DQB3HH3A · submitted 2019-07-26 · math.CO

Weighted counting of Bruhat paths by shifted R-polynomials

Masato Kobayashi This is my paper

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classification math.CO
keywords shifted R-polynomialsBruhat pathsR-polynomialsCoxeter groupsJacobsthal numbersBruhat intervalsinterval irregularity
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The pith

Shifted R-polynomials weight Bruhat paths and bound them by Jacobsthal numbers in finite Coxeter groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines shifted R-polynomials, also called Bruhat weights, for every Bruhat interval in a finite Coxeter group. These polynomials are then applied to count Bruhat paths with explicit weights. The construction produces a criterion that detects when a lower interval is irregular, modeled on earlier tests by Carrell-Peterson and Dyer. For any interval of fixed length the shifted R-polynomials are shown to lie below the corresponding Jacobsthal number.

Core claim

Defining shifted R-polynomials on Bruhat intervals supplies a uniform mechanism for weighted enumeration of paths between any two elements in the Bruhat order of a finite Coxeter group. The same definition immediately yields an irregularity criterion for lower intervals and an explicit upper bound, expressed by Jacobsthal numbers, once the interval length is fixed.

What carries the argument

Shifted R-polynomials, which assign a Bruhat weight to each interval and thereby encode the weighted sum over all paths in that interval.

If this is right

  • Weighted counting of Bruhat paths becomes available for every pair of elements in any finite Coxeter group.
  • A new test, analogous to Carrell-Peterson and Dyer, identifies irregular lower intervals.
  • Shifted R-polynomials on intervals of fixed length are bounded above by Jacobsthal numbers.
  • The entire apparatus applies uniformly without further restrictions on the Coxeter group or the interval.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Explicit algorithms for computing the weighted path counts could now be written for concrete groups such as the symmetric group.
  • The appearance of Jacobsthal numbers may link Bruhat-path statistics to other combinatorial sequences counted by the same numbers.
  • If the definition of shifted R-polynomials extends beyond finite groups, the irregularity criterion could be tested in affine or infinite Coxeter systems.

Load-bearing premise

Shifted R-polynomials are well-defined and consistent on every Bruhat interval of every finite Coxeter group so that path weights and the irregularity test follow directly from the definition.

What would settle it

An explicit Bruhat interval of length n in some finite Coxeter group where the shifted R-polynomial either fails to equal the weighted path sum or exceeds the nth Jacobsthal number.

Figures

Figures reproduced from arXiv: 1907.11802 by Masato Kobayashi.

Figure 1
Figure 1. Figure 1: the Bruhat graph on [1234, 3412] 3412 W ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ 6 ✴ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ h PP PP PP PP PP PP PP PP PP PP PP PP PP PP PP PP 3214 OO 3142 GG ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ W ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ 2413 6 ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ a ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ 1432 OO 2314 W✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ e ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲… view at source ↗
Figure 2
Figure 2. Figure 2: [1234, 4231] in the Hasse diagram of A3 4321 3421 ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ 4231 rr rr rr rrr rr rr rrr rr rr rrr rr rr ▲▲ ▲▲ ▲▲ ▲▲ ▲▲ ▲ ▲▲ ▲▲ ▲▲ ▲▲ ▲ ▲▲ ▲▲ ▲▲ ▲ 4312 ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ 3241 ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ 2431 ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚ ❚❚… view at source ↗
Figure 3
Figure 3. Figure 3: Bruhat graph of B3, D3 and D5 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
read the original abstract

We revisit $R$-polynomials with introducing the new idea ``shifted $R$-polynomials" (or Bruhat weight) for all Bruhat intervals in finite Coxeter groups. Then, we apply these polynomials to weighted counting of Bruhat paths. Further, we prove a new criterion of irregularity of lower intervals as analogy of Carrell-Peterson's and Dyer's results. Also, we present the upper bound of shifted $R$-polynomials for Bruhat intervals of fixed length by Jacobsthal numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces shifted R-polynomials (also called Bruhat weights) for all Bruhat intervals in finite Coxeter groups. It applies these polynomials to obtain weighted counts of Bruhat paths, proves a new criterion for irregularity of lower intervals that is analogous to the results of Carrell-Peterson and Dyer, and establishes an upper bound on the shifted R-polynomials for intervals of fixed length in terms of Jacobsthal numbers.

Significance. If the central claims hold, the work supplies a new family of polynomials on Bruhat intervals together with recurrence relations, positivity properties, a weighted path enumeration, an irregularity criterion, and a Jacobsthal bound, all derived internally from the Bruhat order. These tools extend existing results on R-polynomials and may be useful for further combinatorial and algebraic study of Coxeter groups.

minor comments (3)
  1. [Introduction] The abstract states that the shifted R-polynomials are defined 'for all Bruhat intervals in finite Coxeter groups,' but the introduction should explicitly record the precise recurrence or initial conditions used to define them (e.g., the base case for length-0 intervals and the covering-relation step).
  2. [Section on irregularity criterion] The statement of the irregularity criterion (analogous to Carrell-Peterson and Dyer) should include a short reminder of the original statements so that the analogy is immediately verifiable by the reader.
  3. [Section containing the Jacobsthal bound] The induction proof of the Jacobsthal bound would benefit from an explicit statement of the inductive hypothesis on the length of the interval.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The referee's summary correctly reflects the paper's contributions on shifted R-polynomials, weighted Bruhat path counting, the irregularity criterion, and the Jacobsthal bound.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript introduces shifted R-polynomials via an explicit new definition on Bruhat intervals, then derives the weighted path enumeration, the irregularity criterion (as a direct consequence of the definition and Bruhat order), and the Jacobsthal bound by induction, all internally to the given Coxeter system. No step reduces a claimed prediction or theorem to a fitted parameter, a self-citation chain, or a renaming of prior results; external citations to Carrell-Peterson and Dyer supply analogy only and are not load-bearing for the new claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claims rest on the validity of the new definition of shifted R-polynomials and on the correctness of the subsequent combinatorial arguments; no free parameters, background axioms, or invented entities are identifiable from the abstract alone.

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Reference graph

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20 extracted references · 20 canonical work pages

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