Set partitions without blocks of certain sizes

TL;DR

This paper provides an asymptotic estimate for the number of set partitions avoiding certain block sizes and explores partitions that can be combined into subsets of any size, advancing combinatorial enumeration techniques.

Contribution

It introduces a new asymptotic estimate for partitions with restricted block sizes and analyzes partitions capable of forming all subset sizes, offering novel combinatorial insights.

Findings

Derived asymptotic formulas for restricted block size partitions

Estimated the number of partitions combinable into all subset sizes

Extended enumeration methods to new classes of set partitions

Abstract

We give an asymptotic estimate for the number of partitions of a set of $n$ elements, whose block sizes avoid a given set $\mathcal{S}$ of natural numbers. As an application, we derive an estimate for the number of partitions of a set with $n$ elements, which have the property that its blocks can be combined to form subsets of any size between $1$ and $n$.