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

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 ${\prod }_{k\in \{-n+2\}\cup [-n+4,n-3]\cup \{0\}\cup \{n\}}(x-k)$.

arXiv comment73 pages. This is an extended version of JP's honors thesis (Drexel University 2025), and includes an exposition of the face algebra approach and of the proof of Solomon's Mackey formula. A streamlined research-paper version will be posted separately