Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Arithmetic properties of plane partitions. | Arithmetic properties of plane partitions To Doron a wonderful Mensch Tewodros Amdeberhan Department of Mathematics Tulane University New Orleans LA 70118 tamdeber@ Victor H. Moll Department of Mathematics Tulane University New Orleans LA 70118 vhm@ Submitted Aug 31 2010 Accepted Oct 14 2010 Published Jan 2 2011 Mathematics Subject Classification 05A15 11B75 Abstract The 2-adic valuations of sequences counting the number of alternating sign matrices of size n and the number of totally symmetric plane partitions are shown to be related in a simple manner. Keywords valuations alternating sign matrices totally symmetric plane partitions. 1 Introduction A plane partition PP is an array n nj i j i of nonnegative integers such that n has finite support and is weakly decreasing in rows and columns. These partitions are often represented by solid Young diagrams in 3-dimensions. MacMahon found a complicated formula for the enumeration of all PPs inside an n-cube. This was later simplified to PPn n n i j k 1 1 i j k 1 i j k 2 A plane partition is called symmetric SPP if nij nji for all indices i j. The number of such partitions whose solid Young diagrams fit inside an n-cube is given by i j n 1 i j i 2 sppn nn j 1 i 1 nn j 1 i j i j n 1 i j 1 2 Another interesting subclass of partitions is that of totally symmetric plane partitions TSPP . These are symmetric partitions n such that every row of n is self-conjugate as THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P1 1 an ordinary plane partition or the Young diagrams are invariant under any permutation of the axes . J. Stembridge 3 showed that the number of TSPP in an n-cube is given by n n ni j k 1 _ i j k 2 1 i j k n J i j n 1 y i j n 1 i j i 2 _ i j j 2 u J 1 i j n J J 01 For the solid Young diagram of a plane partition n that fits inside a box of a given size one can take the collection of cubes that are in the box but do not belong to the solid Young diagram. These determine .
