2025/08/08 by Darij Grinberg, Grinberg, Darij, Jonathan Parlett +1
Materials Science · Mathematics · #05E18 #20B30 #60B15 #60J10 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2508.06740
openalex publication_date 2025/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 w0 (which sends each i to n+1-i). More generally, we analyze the composition of any B-basis element of the descent algebra with w0. We show that the minimal polynomial of any such composition (over 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 ∏_k ∈ \set-n+2 ∪ \interval-n+4, n-3 ∪ \set0 ∪ \setn \tupx-k.