Statistical Structure of Concave Compositions

Abstract

In this paper, we study concave compositions, an extension of partitions that were considered by Andrews, Rhoades, and Zwegers. They presented several open problems regarding the statistical structure of concave compositions including the distribution of the perimeter and tilt, the number of summands, and the shape of the graph of a typical concave composition. We present solutions to these problems by applying Fristedt’s conditioning device on the uniform measure.

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Acknowledgements

The authors would like to thank George Andrews, Paweł Hitczenko and Anatoly Vershik for their wonderful insights and helpful comments.

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Correspondence to Amanda Lohss.

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Part of this research was conducted, while the authors were graduate students at Drexel University. Some of the work on this project was funded under European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement No. 335220.

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Dalal, A.J., Lohss, A. & Parry, D. Statistical Structure of Concave Compositions. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00543-6

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Keywords

  • Partitions
  • Concave compositions
  • Limit shape

Mathematics Subject Classification

  • 05A16
  • 60C05
  • 11P82