Probabilistic and Combinatorial Aspects of the Card-Cyclic to Random Insertion Shuffle

TL;DR

This paper investigates a unique card shuffle where each card is removed and reinserted exactly once in order, revealing strong biases and surprising probabilistic and combinatorial properties.

Contribution

It introduces and analyzes a novel nonstandard shuffle process with a single removal and reinsertion per card, uncovering its probabilistic and combinatorial characteristics.

Findings

The shuffle exhibits significant bias in card arrangements.

Surprising combinatorial structures emerge from the process.

The probabilistic analysis reveals strong non-uniformities.

Abstract

Consider a permutation $\sigma\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $\sigma_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We study some probabilistic and combinatorial aspects of the shuffle on $S_n$ defined by removing and then randomly reinserting each of the $n$ cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be quite strong and possesses some surprising features.