TAILIEUCHUNG - Báo cáo toán học: "(−1)–enumeration of self–complementary plane partitions"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: (−1)–enumeration of self–complementary plane partitions. | 1 -enumeration of self-complementary plane partitions Theresia Eisenkolbl Fakultat fur Mathematik Universitat Wien Nordbergstrafie 15 A-1090 Wien Austria. Submitted Dec 15 2004 Accepted Jan 4 2005 Published Jan 11 2005 Mathematics Subject Classifications 05A15 05B45 52C20 Abstract We prove a product formula for the remaining cases of the weighted enumeration of self-complementary plane partitions contained in a given box where adding one half of an orbit of cubes and removing the other half of the orbit changes the sign of the weight. We use nonintersecting lattice path families to express this enumeration as a Pfaffian which can be expressed in terms of the known ordinary enumeration of self-complementary plane partitions. 1 Introduction A plane partition P can be defined as a finite set of points i j k with i j k 0 and if i j k G P and 1 i i 1 j j 1 k k then i j k 2 P. We interpret these points as midpoints of cubes and represent a plane partition by stacks of cubes see Figure 1 . If we have i a j b and k c for all cubes of the plane partition we say that the plane partition is contained in a box with sidelengths a b c. Plane partitions were first introduced by MacMahon. One of his main results is the following 10 Art. 429 x 1 proof in Art. 494 The number of all plane partitions contained in a box with sidelengths a b c equals a Y c B a b c nnn i 1j 1k 1 A c i b U i b i 1 1 i j k 1 i j k 2 where a n a a 1 a 2 . a n 1 is the rising factorial. MacMahon also started the investigation of the number of plane partitions with certain symmetries in a given box. These numbers can also be expressed as product formulas similar to the one given above. In 14 Stanley introduced additional complementation THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R7 1 Figure 1 A self-complementary plane partition symmetries giving six new combinations of symmetries which led to more conjectures all of which were settled in the 1980 s and 90 s see 14 8 3

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