## Introduction

The partitioning of biodiversity into different spatial components is critical to understand processes underlying species distributions and diversity turnover (Magurran 2004; Ackerly & Cornwell 2007; Prinzing et al. 2008; de Bello et al. 2009). In particular, proper management of ecosystems requires that we understand the processes by which β-diversity (i.e. the diversity across habitats or communities) is generated and maintained (Legendre et al. 2005). Several indices and mathematical frameworks have been developed for these purposes (Lande 1996; Veech et al. 2002; Crist & Veech 2006), making it possible to answer different ecological questions and unavoidably producing little consensus among methods (Koleff et al. 2003). Overall, it is widely accepted that the total diversity of a region (γ-diversity) can be partitioned into within-community (α-diversity) and among-communities (β-diversity; Whittaker 1975; Magurran 2004 and references therein) components. Partitioning of diversity could then be additive (e.g. γ=α+β) or multiplicative (γ=α^{*}β), depending on the models and mathematical indices used (Veech et al. 2002; Ricotta 2005a; Jost 2007; Jost et al. 2010).

Among the different existing mathematical frameworks (Magurran 2004), Rao's quadratic entropy index (1982) can provide a general approach for partitioning biodiversity into α, β and γ components. Indeed, the Rao entropy is currently the only existing estimator of diversity that formally combines different measures of species dissimilarity (e.g. phylogenetic or functional) with relative species abundances, providing a standardized methodology applicable to compare α, β and γ components between different facets of diversity (e.g. taxonomic, phylogenetic and functional diversity; Pavoine et al. 2004; Ricotta 2005a; Hardy & Senterre 2007). Furthermore, the index provides one of the few direct measures of species redundancy within and among biological communities (de Bello et al. 2007, 2009). These unique properties of the Rao index could open new perspectives to understand mechanisms driving the turnover of diversity along environmental and temporal scales.

However, some key methodological issues regarding the spatial partitioning of diversity with the Rao index have been hotly debated in the recent literature (Ricotta 2005a, b; Hardy & Jost 2008; Villeger & Mouillot 2008; de Bello et al. 2009). Whether the index could lead to negative β values has been discussed (Hardy & Jost 2008; Villeger & Mouillot 2008), with no clear agreement yet on how Rao's index should be computed. Moreover, the index has not been able to offer a robust estimation of β-diversity (de Bello et al. 2009), giving systematically low estimates of β-diversity, even for complete species replacement between communities. To help ecologists find a way out of this mathematical labyrinth, we provide clarification and technical guideline to derive a more realistic partition of diversity using the Rao index (i.e. producing a β-diversity that behaves as ecologists would expect). We discuss these issues with numerical examples and a case study, which demonstrate how promising a corrected version of the Rao index for the partitioning of α, β and γ components of diversity can be. As the idea for the study was conceived during the IAVS meeting held in Crete (2009), we hope this study will constitute a thread to follow for ecologists, as in the Theseus myth, and revive classic ecological questions using a reliable mathematical framework.