Integer Partitions Probability Distributions

TL;DR

This paper introduces two related discrete probability distributions based on integer partitions, analyzing their properties and deriving expectations and covariances, with conjectures for some elements of the expectation vector.

Contribution

It presents new distributions derived from integer partitions and explores their statistical properties, including expectations, covariances, and related conjectures.

Findings

Derived expectation vector and covariance matrix for one distribution.

Provided conjectures for elements of the expectation vector for the other distribution.

Distributions are connected to permutations of n letters.

Abstract

Two closely related discrete probability distributions are introduced. In each case the support is a set of vectors in $\mathbb{R}^n$ obtained from the partitions of the fixed positive integer $n$. These distributions arise naturally when considering equally-likely random permutations on the set of $n$ letters. For one of the distributions, the expectation vector and covariance matrix is derived. For the other distribution, conjectures for several elements of the expectation vector are provided.