Tamari Lattices and the symmetric Thompson monoid

We investigate the connection between Tamari lattices and the Thompson group F, summarized in the fact that F is a group of fractions for a certain monoid F+sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F+sym. As an application, we show that, for every n, there exists a length l chain in the nth Tamari lattice whose endpoints are at distance at most 12l/n.Comment: 35page

“An important cog in the wheel”, but not the driver: Coaches’ perceptions of their role in doping prevention

Objectives: Under the World Anti-Doping Code coaches have designated anti-doping roles and responsibilities. Yet, their experiences, opinions and behaviours in relation to these expectations are poorly understood. This study responds directly to this absence of evidence in order to move the field forward. Design: A qualitative thematic analysis approach. Method: Twelve football and rugby league coaches, working in a performance development context, took part in semi-structured interviews to explore their (anti-)doping experiences, opinions and behaviours. Nine coaches participated in follow-up interviews where particular attention was paid to existing anti-doping policy directives. All interviews were analysed using inductive thematic analysis. Results: Coaches were supportive of anti-doping efforts and exerted their influence by monitoring, giving advice and creating the ‘right’ culture. Performance prioritisation rendered coaches reluctant to engage proactively in addressing anti-doping in their practice; a situation exacerbated by a lack of self-efficacy to advise/act in accordance with the rules. Consequently, coaches tended to rely on others (both internally and externally to their club) to provide anti-doping support, and anti-doping is deemed unnecessary/irrelevant. Critically, coaches’ current behaviours were not driven by policy, as they were unaware of expectations and consequences outlined in the Code. Conclusions: Coaches are willing to support anti-doping efforts, but are generally passive in their everyday practice. The gulf between anti-doping policy and coaching practice raises cause for concern for anti-doping policy makers. To bridge this gap systematic programming of activities designed to ensure coaches are able and willing to take a proactive role in doping prevention is required