# Refined Cyclic Sieving on Words for the Major Index Statistic

## TL;DR

This paper introduces a combinatorial refinement of the cyclic sieving phenomenon related to the major index statistic on words with fixed content and cyclic descent type, providing a new bijective proof approach.

## Contribution

It formulates and proves a combinatorial refinement of CSP for the major index statistic, differing from previous representation-theoretic methods, and introduces the universal sieving statistic "flex".

## Key findings

- Refined CSP for the major index on words with fixed content.
- A new combinatorial, bijective proof approach.
- Extension of cyclic sieving to shifted subset sums.

## Abstract

Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content. There is a natural notion of refinement for many CSP's. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argument presented is completely different from Reiner-Stanton-White's representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective in a sense we make precise with a "universal" sieving statistic on words, "flex". A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner-Stanton-White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon-Wilf.