Variance of additive functions defined on random assemblies

TL;DR

This paper establishes a variance inequality for additive functions on random assemblies, generalizing previous results for permutations and mappings, and drawing an analogy to the Turán-Kubilius inequality in number theory.

Contribution

It introduces a new variance inequality for additive functions on assemblies, extending known results to a broader class of combinatorial structures.

Findings

Variance inequality for additive functions on assemblies

Generalization of permutation and mapping results

Analogy to Turán-Kubilius inequality in number theory

Abstract

An inequality for the variance of an additive function defined on random decomposable structures, called assemblies, is established. The result generalizes estimates obtained earlier in the cases of permutations and mappings of a finite set into itself. It is analogous to the Tur\'an-Kubilius inequality for additive number-theoretic functions.