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Multiplicity of summands in the random partitions of an integer

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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 ~An α 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 \(\alpha = \frac{1}{3}\).

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Acknowledgements

The author would like to thank Professor Rajesh Ravindran for introducing him to the subject of integer partitions. He would also like to thank Professor Rahul Roy and the referee for their crucial comments.

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Authors and Affiliations

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

    Ghurumuruhan Ganesan

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  1. Ghurumuruhan Ganesan
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Correspondence to Ghurumuruhan Ganesan.

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Ganesan, G. Multiplicity of summands in the random partitions of an integer. Proc Math Sci 123, 101–143 (2013). https://doi.org/10.1007/s12044-012-0107-2

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  • DOI: https://doi.org/10.1007/s12044-012-0107-2

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