# Weighted counting of Bruhat paths by shifted $R$-polynomials

## TL;DR

This paper introduces shifted R-polynomials for finite Coxeter groups, applying them to weighted Bruhat path counting, establishing irregularity criteria, and providing bounds related to Jacobsthal numbers.

## Contribution

It proposes shifted R-polynomials for all Bruhat intervals, offering new tools for weighted path counting and irregularity analysis in Coxeter groups.

## Key findings

- Introduces shifted R-polynomials as Bruhat weights.
- Provides a new irregularity criterion for lower intervals.
- Establishes upper bounds of shifted R-polynomials using Jacobsthal numbers.

## 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.