Real Zeros and Partitions without singleton blocks

TL;DR

This paper proves that the generating polynomials for partitions of an n-element set into non-singleton blocks have only real roots and analyzes their asymptotic behavior to determine the most probable number of blocks.

Contribution

It establishes the real-rootedness of generating polynomials for specific set partitions and studies their asymptotic properties, providing new insights into the distribution of the number of blocks.

Findings

Generating polynomials have only real roots.

Asymptotic behavior of the leftmost roots is characterized.

Most likely number of blocks is determined from root analysis.

Abstract

We prove that the generating polynomials of partitions of an $n$-element set into non-singleton blocks, counted by the number of blocks, have real roots only and we study the asymptotic behavior of the leftmost roots. We apply this information to find the most likely number of blocks.