# The recombination equation for interval partitions

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## 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 (Baake et al. in Discr Cont Dynam Syst 36:63–95, 2016) 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.

## Keywords

Recombination equation Population genetics Markov generator Interval partitions Measure-valued equations Nonlinear ODEs Closed solution## Mathematics Subject Classification

34G20 06B23 92D10 60J25## Notes

### Acknowledgements

It is our pleasure to thank E. Baake and M. Salamat for valuable discussions. ES is grateful to the FSPM\(^2\) of Bielefeld University for financial support during her stay in Bielefeld. This work was also supported by the German Research Foundation (DFG), within the SPP 1590.

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