Top to random and reverse: analysis of a new descent algebra shuffle

TL;DR

This paper investigates a new type of shuffle combining top-to-random and reversal operations, analyzing its algebraic properties and minimal polynomial structure within the descent algebra framework.

Contribution

It introduces the analysis of the top-to-random-and-reverse shuffle and characterizes the minimal polynomial of related algebraic elements, revealing their factorization properties.

Findings

Minimal polynomial factors into distinct linear factors.

Minimal polynomial explicitly given for the top-to-random-and-reverse shuffle.

Signed knapsack numbers relate to the polynomial's roots.

Abstract

We study the "top-to-random-and-reverse shuffle", defined as the top-to-random shuffle in the symmetric group algebra composed with the permutation $w_0$ (which sends each $i$ to $n+1-i$). More generally, we analyze the composition of any B-basis element of the descent algebra with $w_0$. We show that the minimal polynomial of any such composition (over $\mathbb{Q}$) factors into distinct linear factors, which correspond to the "signed knapsack numbers" of set compositions. This is a counterpart to an analogous property of the B-basis elements themselves, which was proved by Brown using Bidigare's face monoid. In the case of the top-to-random-and-reverse shuffle, the minimal polynomial turns out to be $\prod_{k \in \set{-n+2} \cup \interval{-n+4, n-3} \cup \set{0} \cup \set{n}} \tup{x-k}$.