Determinantal Formula for Generalized Riffle Shuffle

TL;DR

This paper introduces a determinantal formula for calculating the probability of descents in a generalized riffle shuffle on colored permutation groups, using a bijection with non-intersecting lattice paths.

Contribution

It provides a novel determinantal expression for descent probabilities in a generalized shuffle model, linking combinatorics with lattice path bijections.

Findings

Determinantal formula for descent probabilities

Bijection between shuffles and non-intersecting lattice paths

Analytical tool for generalized riffle shuffles

Abstract

We consider a generalized riffle shuffle on the colored permutation group $G_{p, n}$ and derive a determinantal formula for the probability of finding descents at given positions, proof of which is based on the bijection between the set of shuffles in question and that of non-intersecting lattice paths.