We study the distribution of the number of parts of given multiplicity (or equivalently, ascents of given size) in integer partitions. In this paper we give methods to compute asymptotic formulas for the expected value and variance of the number of parts of multiplicity d (d is a positive integer) in a random partition of a large integer n and also prove that the limiting distribution is asymptotically normal for fixed d. However, if we let d increase with n, we get a phase transition for d around n 1/4. Our methods can also be applied to the so−called λ-partitions where the parts are members of a sequence of integers λ.
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Ralaivaosaona, D. On the Distribution of Multiplicities in Integer Partitions. Ann. Comb. 16, 871–889 (2012). https://doi.org/10.1007/s00026-012-0165-2
Mathematics Subject Classification
- integer partitions
- asymptotic expansions
- limit distribution