| Diagonal Sums of Boxed Plane Partitions (2007) | |||||||||||||
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| Abstract: We give a simple proof of a nice formula for the means and covariances of the diagonal sums of a uniformly random boxed plane parition. An a \Theta b \Theta c boxed plane partition is an a \Theta b grid of integers between 0 and c inclusive, such that the numbers decrease weakly in each row and column. At the right is a 4 \Theta 5 \Theta 6 boxed plane parition, which for convenience we have drawn rotated 45 ffi. We have added up these numbers in the direction along the main diagonal of the lattice to obtain the diagonal sums S \Gammaa+1; : : : ; S b\Gamma1. If we pick the boxed plane partition uniformly at random, these form a sequence of random variables, and we show that their means and covariances are given by 6 | |||||||||||||
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