| Diagonal Sums of Boxed Plane Partitions (2007) | |||||||||||||
Abstract | |||||||||||||
| 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 × b × c boxed plane partition is an a × 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 × 5 × 6 boxed plane parition, which for convenience we have drawn rotated 45 ◦. We have added up these numbers in the direction along the main diagonal of the lattice to obtain the diagonal sums S−a+1,...,Sb−1. 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 (a + i)bc/(a + b) i ≤ 0 E[Si]= (b − i)ac/(a + b) i ≥ 0 Cov(Si,Sj)=(a + i)(b − j) × abc(a + b + c) (a + b) 2 ((a + b) 2 − 1) 4 5 | |||||||||||||
Publication details | |||||||||||||
| |||||||||||||