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Báo cáo toán học: "Counting k-Marked Durfee Symbols"

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: Counting k-Marked Durfee Symbols. | Counting k-Marked Durfee Symbols Kagan Kur ungoz Department of Mathematics The Pennsylvania State University University Park PA 16802 kursun@math.psu.edu Submitted May 7 2010 Accepted Feb 5 2011 Published Feb 14 2011 Mathematics Subject Classification 05A15 05A05 Abstract An alternative characterization of k-marked Durfee symbols defined by Andrews is given. Some identities involving generating functions of k-marked Durfee symbols are proven combinatorially by considering the symbols not individually but in equivalence classes. Also a related binomial coefficient identity is obtained in the course. A partition A of a positive integer n is a nonincreasing sequence of positive integers Al Ak 0 such that n Al Ak 1 Ch.1 . A pictorial representation for a partition is its Ferrers graph a left indented table of dots such that the first row has A1 dots and so on. For instance 36 9 7 7 5 4 2 2 has the following Ferrers graph. The largest square that can be fit in the upper left corner of a Ferrers graph is called the Durfee square. So the partition above has a Durfee square of side length 4 . The rank of a partition is defined as the largest part minus the number of parts in a partition 5 . The rank of the above partition is 2. The conjugate A of a partition A is obtained by reflecting the Ferrers graph across its main diagonal. For the partition above the conjugate is 7 7 5 5 4 3 3 1 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P41 1 and has the following Ferrers graph. Using the Ferrers graph of a partition we can form another representation of the same partition the Durfee symbol 3 . It is a two-row array with a subscript indicating the side length of the Durfee square. The top row is obtained by reading the conjugate partition of the smaller partition to the right of the Durfee square recording the columns instead of rows and the bottom row by reading the smaller partition below the Durfee square. Thus the partition 36 9 7 7 5 4 2 2 has Durfee symbol 4 3 3 1 1 422