Abstract Given a permutation statistic$$\operatorname {\mathrm {st}}$$, define its inverse statistic$$\operatorname {\mathrm {ist}}$$by. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$whenever$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$. Our work leads to a rederivation of Stanley’s generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and$$\gamma $$-positivity. 
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                            Measuring the stellar and planetary parameters of the 51 Eridani system
                        
                    
    
            Abstract In order to study exoplanets, a comprehensive characterisation of the fundamental properties of the host stars – such as angular diameter, temperature, luminosity, and age, is essential, as the formation and evolution of exoplanets are directly influenced by the host stars at various points in time. In this paper, we present interferometric observations taken of directly imaged planet host 51 Eridani at the CHARA Array. We measure the limb-darkened angular diameter of 51 Eridani to be$$\theta_\mathrm{LD} = 0.450\pm 0.006$$mas and combining with the Gaia zero-point corrected parallax, we get a stellar radius of$$1.45 \pm 0.02$$R$$_{\odot}$$. We use the PARSEC isochrones to estimate an age of$$23.2^{+1.7}_{-2.0}$$Myr and a mass of$$1.550^{+0.006}_{-0.005}$$M$$_{\odot}$$. The age and mass agree well with values in the literature, determined through a variety of methods ranging from dynamical age trace-backs to lithium depletion boundary methods. We derive a mass of$$4.1\pm0.4$$M$$_\mathrm{Jup}$$for 51 Eri b using the Sonora Bobcat models, which further supports the possibility of 51 Eri b forming under either the hot-start formation model or the warm-start formation model. 
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                            - Award ID(s):
- 2034336
- PAR ID:
- 10569148
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Publications of the Astronomical Society of Australia
- Volume:
- 41
- ISSN:
- 1323-3580
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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