• Multiplicity of Summands in the Random Partitions of an Integer

• # Fulltext

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 $A&gt;0$ and $\alpha &lt; \frac{1}{2}$. 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

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

• # Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019