Generalized Galois numbers, inversions, lattice paths, Ferrers diagrams and limit theorems

TL;DR

This paper provides probabilistic interpretations of generalized Galois numbers through inversions, lattice paths, and Ferrers diagrams, leading to new proofs and results on their limit behaviors.

Contribution

It introduces novel probabilistic perspectives on generalized Galois numbers, connecting them to combinatorial structures and deriving new limit theorems.

Findings

Probabilistic interpretations via inversions, lattice paths, Ferrers diagrams

New proofs of existing limit theorems for Galois numbers

Additional limit results derived from combinatorial models

Abstract

Bliem and Kousidis (arXiv:1109.4624) recently considered a family of random variables whose distributions are given by the generalized Galois numbers (after normalization). We give probabilistic interpretations of these random variables, using inversions in random words, random lattice paths and random Ferrers diagrams, and use these to give new proofs of limit theorems as well as some further limit results.