Unified generating function for set partitions

TL;DR

This paper introduces a unified generating function for nine types of set partitions, exploring its properties, combinatorial explanations, and new statistics, with explicit formulas for related polynomials.

Contribution

It presents a novel unified generating function for various set partitions and provides combinatorial insights and explicit formulas for associated polynomials.

Findings

Unified generating function for 9 types of set partitions

New combinatorial statistics defined and analyzed

Explicit formulas derived for polynomial coefficients

Abstract

In this work we define a unified generating functions for 9 different kinds of set partitions including cyclically ordered set partitions. Such generating function depends on 4 parameters. We consider property of this function and provide combinatorial explanation for polynomials generated by this function. Two new combinatorial statistics are defined and the explicit formulae given for coefficients of parametrized polynomials defined by the generating function.