Statistical Structure of Concave Compositions


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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  1. 1.

    Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1998). Reprint of the 1976 original

  2. 2.

    Andrews, G.E.: Concave compositions. Electron. J. Combin. 18(2), Paper 6, 13 (2011)

  3. 3.

    Andrews, G.E.: Concave and convex compositions. Ramanujan J. 31(1-2), 67–82 (2013).

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Andrews, G.E., Rhoades, R.C., Zwegers, S.P.: Modularity of the concave composition generating function. Algebra Number Theory 7(9), 2103–2139 (2013).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bender, E.A., Canfield, E.R.: Locally restricted compositions. I. Restricted adjacent differences. Electron. J. Combin. 12, Research Paper 57, 27 pp. (electronic) (2005).

  6. 6.

    Bryson, J., Ono, K., Pitman, S., Rhoades, R.C.: Unimodal sequences and quantum and mock modular forms. Proc. Natl. Acad. Sci. USA 109(40), 16063–16067 (2012).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Chaganty, N.R., Sethuraman, J.: Strong large deviation and local limit theorems. Ann. Probab. 21(3), 1671–1690 (1993).<1671:SLDALL>2.0.CO;2-2&origin=MSN

  8. 8.

    Corteel, S., Pittel, B., Savage, C.D., Wilf, H.S.: On the multiplicity of parts in a random partition. Random Structures Algorithms 14(2), 185–197 (1999).<185::AID-RSA4>3.3.CO;2-6.

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Fristedt, B.: The structure of random partitions of large integers. Trans. Amer. Math. Soc. 337(2), 703–735 (1993).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Goh, W.M.Y., Hitczenko, P.: Average number of distinct part sizes in a random Carlitz composition. European J. Combin. 23(6), 647–657 (2002).

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Goh, W.M.Y., Hitczenko, P.: Random partitions with restricted part sizes. Random Structures Algorithms 32(4), 440–462 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Grabner, P., Knopfmacher, A., Wagner, S.: A general asymptotic scheme for moments of partition statistics. preprint (2010)

  13. 13.

    Heubach, S., Mansour, T.: Combinatorics of Compositions and Words. Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL (2010)

  14. 14.

    Loève, M.: Probability Theory. Third edition. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London (1963)

  15. 15.

    MacMahon, P.A.: Combinatory Analysis. Vol. I, II (bound in one volume). Dover Phoenix Editions. Dover Publications, Inc., Mineola, NY (2004). Reprint of ıt An introduction to combinatory analysis (1920) and ıt Combinatory analysis. Vol. I, II (1915, 1916)

  16. 16.

    Ngo, T.H., Rhoades, R.C.: Integer partitions, probabilities and quantum modular forms. preprint (2014)

  17. 17.

    Petrov, F.: Two elementary approaches to the limit shapes of Young diagrams. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 370(Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 40), 111–131, 221 (2009).

  18. 18.

    Szalay, M., Turán, P.: On some problems of the statistical theory of partitions with application to characters of the symmetric group. I. Acta Math. Acad. Sci. Hungar. 29(3-4), 361–379 (1977)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Temperley, H.N.V.: Statistical mechanics and the partition of numbers. II. The form of crystal surfaces. Proc. Cambridge Philos. Soc. 48, 683–697 (1952)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Vershik, A.M.: Asymptotic combinatorics and algebraic analysis. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 1384–1394. Birkhäuser, Basel (1995)

  21. 21.

    Vershik, A.M.: Statistical mechanics of combinatorial partitions, and their limit configurations. Funktsional. Anal. i Prilozhen. 30(2), 19–39, 96 (1996).

  22. 22.

    Vershik, A.M., Kerov, S.V.: Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group. Funktsional. Anal. i Prilozhen. 19(1), 25–36, 96 (1985)

  23. 23.

    Wright, E.M.: Stacks. Quart. J. Math. Oxford Ser. (2) 19, 313–320 (1968)

    Article  Google Scholar 

  24. 24.

    Wright, E.M.: Stacks. II. Quart. J. Math. Oxford Ser. (2) 22, 107–116 (1971)

    Article  Google Scholar 

  25. 25.

    Wright, E.M.: Stacks. III. Quart. J. Math. Oxford Ser. (2) 23, 153–158 (1972)

    Article  Google Scholar 

  26. 26.

    Yakubovich, Y.: Ergodicity of multiplicative statistics. J. Combin. Theory Ser. A 119(6), 1250–1279 (2012).

    MathSciNet  Article  MATH  Google Scholar 

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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).

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  • Partitions
  • Concave compositions
  • Limit shape

Mathematics Subject Classification

  • 05A16
  • 60C05
  • 11P82