The limiting distribution of the trace of a random plane partition |
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| Authors: | E P Kamenov L R Mutafchiev |
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| Institution: | (1) Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, James Bouchier Blvd. 5, Sofia, 1164, Bulgaria;(2) American University in Bulgaria, 2700 Blagoevgrad, Bulgaria;(3) Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, Acad. G. Bonche str., bl. 8, 1113 Sofia, Bulgaria |
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| Abstract: | We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ
n
= τ
n
(ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ
n
− c
0
n
2/3)/c
1
n
1/3 log1/2
n converges weakly, as n → ∞, to the standard normal distribution, where c
0 = ζ(2)/ 2ζ(3)]2/3, c
1 = √(1/3/) 2ζ(3)]1/3 and ζ(s) = Σ
j=1∞
j
−s
.
Partial support given by the National Science Fund of the Bulgarian Ministry of Education and Science, grant No. VU-MI-105/2005. |
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| Keywords: | trace of a plane partition central limit theorems |
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