Involutions and their progenies

TL;DR

This paper studies permutations with specific cycle structures, especially involutions, providing formulas, asymptotic analysis, and properties to understand their combinatorial behavior.

Contribution

It introduces new generating functions and determinantal formulas for involutions and their partial sums, advancing the understanding of their combinatorial and arithmetic properties.

Findings

Derived generating functions for involutions

Established asymptotic estimates for cycle-type permutations

Explored arithmetic and combinatorial properties of involutions

Abstract

Any permutation has a disjoint cycle decomposition and concept generates an equivalence class on the symmetry group called the cycle-type. The main focus of this work is on permutations of restricted cycle-types, with particular emphasis on the special class of involutions and their partial sums. The paper provides generating functions, determinantal expressions, asymptotic estimates as well as arithmetic and combinatorial properties.