Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

TL;DR

This paper introduces the concept of asymptotic freedom in locally restricted compositions, showing large parts behave like independent Poisson variables, and derives precise asymptotic probabilities and expectations for gap-free compositions.

Contribution

It defines asymptotically free compositions and establishes their large parts follow asymptotically geometric distributions, leading to new asymptotic formulas for various composition parameters.

Findings

Large parts follow asymptotically geometric distributions.

Number of large parts are asymptotically independent Poisson variables.

Derived precise asymptotic probabilities for gap-free compositions.

Abstract

We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part, number of distinct parts and number of parts of multiplicity k, all accurate to o(1).