## Graduation Year

2014

## Document Type

Open Access Senior Thesis

## Degree Name

Bachelor of Science

## Department

Mathematics

## Reader 1

Nicholas Pippenger

## Reader 2

Michael Orrison

## Terms of Use & License Information

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

## Rights Information

© 2014 Tongjia Shi

## Abstract

Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θ^{K}, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The m^{th} moment of the r^{th} longest cycle of a random permutation is Θ(n^{m}), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or negatively correlated, depending on θ. The m^{th} moments of the r^{th} shortest cycle of a random permutation is Θ(n^{m−θ}/(ln n)^{r−1}) when θ < m, Θ((ln n)^{r}) when θ = m, and Θ(1) when θ > m. The exponent of cycle lengths at the 100q^{th} percentile goes to q with zero variance. The exponent of the expected cycle lengths at the 100q^{th} percentile is at least q due to the Jensen’s inequality, and the exact value is derived.

## Recommended Citation

Shi, Tongjia, "Cycle lengths of θ-biased random permutations" (2014). *HMC Senior Theses*. 65.

https://scholarship.claremont.edu/hmc_theses/65

## Source Fulltext

http://www.math.hmc.edu/~tshi/thesis/permutations.pdf