Riffles, ruffles, and the turning algebra

TL;DR

This paper introduces new subalgebras related to riffle shuffles, called ruffles, which involve turning over the bottom cards after cutting, expanding the algebraic framework of card shuffling processes.

Contribution

It proposes novel subalgebras associated with ruffles, extending the algebraic structures used to analyze card shuffling beyond the rising algebra.

Findings

Defines new subalgebras related to ruffles

Connects ruffles to algebraic structures in symmetric groups

Provides foundational concepts for future research

Abstract

The rising algebra is a subalgebra of the group algebra of the symmetric group S_n, gotten by lumping together permutations having the same number of rising sequences. This well-known algebra arises naturally when studying riffle shuffles. Here we introduce a number of other subalgebras that arise naturally when studying `ruffles', which are like riffles except that after cutting the deck you turn over the bunch of cards that were on the bottom. This orphaned draft offers no context or motivation, and uses idiosyncratic notation and terminology that `seemed like a good idea at the time'. We're making it available because it has been cited in this form.