The recombination equation for interval partitions

TL;DR

This paper analyzes the deterministic recombination equation on the lattice of interval partitions, deriving explicit solutions and examining their decay properties and recursive structure, with implications for understanding partitioning processes.

Contribution

It provides a detailed solution for the recombination equation on interval partitions, extending the general framework to this specific lattice and analyzing decay rates and recursive structures.

Findings

Explicit solution for interval partition recombination equation

Analysis of decay rates and recursive structure

Insights into partitioning process dynamics

Abstract

The general deterministic recombination equation in continuous time is analysed for various lattices, with special emphasis on the lattice of interval (or ordered) partitions. Based on the recently constructed general solution for the lattice of all partitions, the corresponding solution for interval partitions is derived and analysed in detail. We focus our attention on the recursive structure of the solution and its decay rates, and also discuss the solution in the degenerate cases, where it comprises products of monomials with exponentially decaying factors. This can be understood via the Markov generator of the underlying partitioning process that was recently identified. We use interval partitions to gain insight into the structure of the solution, while our general framework works for arbitrary lattices.