TAILIEUCHUNG - Báo cáo toán học: "Diagonal Sums of Boxed Plane Partitions"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Diagonal Sums of Boxed Plane Partitions. | Diagonal Sums of Boxed Plane Partitions David B. Wilson Microsoft Research One Microsoft Way Redmond WA 98052 dbwilson@ Submitted December 8 2000 Accepted December 22 2000 2000 AMS subject classifications primary 60C05 secondary 05A15. 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 X b X c boxed plane partition is an a X 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 X 5 X 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 4 8 10 14 9 6 4 1 Figure 1 A 4 X 5 X 6 boxed plane partition with its contours and diagonal sums. E Si J a i bc a b i b i ac a b i 0 i 0 Í t . abc a b c . - A Cov S- Sj a i b j X a bfi a b 2 1 i j Notice that while the expected values see the corner at the origin the other corner does not enter the formula and neither corner enters the formula for the covariances. For given values of a b and c the covariances are just proportional to the product of the distances from i and j to the endpoints. As Kenyon points out a similar covariance property holds for Brownian bridges and indeed can be deduced from this formula by taking c 1 and a b 1. We are unaware of similarly nice formulas for . the row sums. As Stembridge points out diagonal sums appear in some generating functions such as 0 b-1 E. .IK 01 xxL-x i i 1 j a X b X 1 bpp s i i -a 1 j 0 J THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 N1 1 which is due to Stanley see 3 Chapter 7 . The corresponding generating function for a X b X c bpp s is not so nice but Krattenthaler 2 pp 192 expresses it in terms

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