• Multiplicity of Summands in the Random Partitions of an Integer

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      https://www.ias.ac.in/article/fulltext/pmsc/123/01/0101-0143

    • Keywords

       

      Yakubovich conjecture; repeated summands; slices of Young diagrams.

    • Abstract

       

      In this paper, we prove a conjecture of Yakubovich regarding limit shapes of `slices’ of two-dimensional (2D) integer partitions and compositions of 𝑛 when the number of summands $m\sim An^\alpha$ for some $A>0$ and $\alpha < \frac{1}{2}$. We prove that the probability that there is a summand of multiplicity 𝑗 in any randomly chosen partition or composition of an integer 𝑛 goes to zero asymptotically with 𝑛 provided 𝑗 is larger than a critical value. As a corollary, we strengthen a result due to Erdös and Lehner (Duke Math. J. 8(1941) 335–345) that concerns the relation between the number of integer partitions and compositions when $\alpha=\frac{1}{3}$.

    • Author Affiliations

       

      Ghurumuruhan Ganesan1

      1. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi 110 016, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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