Set partitions with no m-nesting

TL;DR

This paper studies set partitions avoiding m-nestings, using generating trees and algebraic methods to derive functional equations and analyze the complexity of their enumeration.

Contribution

It introduces a new combinatorial construction for m-nesting-free partitions and develops algebraic tools to derive their generating functions.

Findings

Derived functional equations for partitions with no m-nesting.

Applied algebraic kernel method for coefficient extraction.

Showed increasing complexity of formulas as m grows.

Abstract

A partition on [n] has an m-nesting if there exists i_1 < i_2 < ... < i_m < j_m < j_{m-1} < ... < j_1, where i_l and j_l are in the same block for all 1 <= l <= m. We use generating trees to construct the class of partitions with no m-nesting and determine functional equations satisfied by the associated generating functions. We use algebraic kernel method together with a linear operator to describe a coefficient extraction process. This gives rise to enumerative data, and illustrates the increasing complexity of the coefficient formulas as m increases.