# Sum of weighted records in set partitions

## TL;DR

This paper derives an explicit formula and asymptotic estimates for the total sum of weighted records in set partitions, linking it to Bell numbers and analyzing its generating function.

## Contribution

It introduces a new explicit formula and asymptotic analysis for the sum of weighted records in set partitions, expanding understanding of this combinatorial statistic.

## Key findings

- Derived explicit formula for sum of weighted records
- Provided asymptotic estimates related to Bell numbers
- Analyzed the generating function for the statistic

## Abstract

The purpose of this paper is to find an explicit formula and asymptotic estimate for the total number of sum of weighted records over set partitions of $[n]$ in terms of Bell numbers. For that we study the generating function for the number of set partitions of $[n]$ according to the statistic sum of weighted records.