On the distribution of the longest run in number partitions |
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Authors: | Stephan Wagner |
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Institution: | (1) American University in Bulgaria, 2700 Blagoevgrad, Bulgaria E-mail |
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Abstract: | We consider the distribution of the longest run of equal elements in number partitions (equivalently, the distribution of
the largest gap between subsequent elements); in a recent paper, Mutafchiev proved that the distribution of this random variable
(appropriately rescaled) converges weakly. The corresponding distribution function is closely related to the generating function
for number partitions. In this paper, this problem is considered in more detail—we study the behavior at the tails (especially
the case that the longest run is comparatively small) and extend the asymptotics for the distribution function to the entire
interval of possible values. Additionally, we prove a local limit theorem within a suitable region, i.e. when the longest
run attains its typical order n
1/2, and we observe another phase transition that occurs when the largest gap is of order n
1/4: there, the conditional probability that the longest run has length d, given that it is ≤d, jumps from 1 to 0. Asymptotics for the mean and variance follow immediately from our considerations. |
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Keywords: | |
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