TAILIEUCHUNG - Báo cáo toán học: "Shift-Induced Dynamical Systems on Partitions and Compositions Brian Hopkins"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Shift-Induced Dynamical Systems on Partitions and Compositions Brian Hopkins. | Shift-Induced Dynamical Systems on Partitions and Compositions Brian Hopkins Department of Mathematics Saint Peter s College Jersey City NJ 07306 USA bhopkins@ Michael A. Jones Department of Mathematical Sciences Montclair State University Montclair NJ 07043 jonesm@ Submitted Feb 22 2006 Accepted Sep 10 2006 Published Sep 22 2006 Mathematics Subject Classification 05A17 37E15 Abstract The rules of Bulgarian solitaire are considered as an operation on the set of partitions to induce a finite dynamical system. We focus on partitions with no preimage under this operation known as Garden of Eden points and their relation to the partitions that are in cycles. These are the partitions of interest as we show that starting from the Garden of Eden points leads through the entire dynamical system to all cycle partitions. A primary result concerns the number of Garden of Eden partitions the number of cycle partitions is known from Brandt . The same operation and questions can be put in the context of compositions ordered partitions where we give stronger results. 1 Introduction Let P n be the set of partitions of n. The relation A 2 P n will be written A n. The shift operator DP P n P n is defined as follows. Given a partition A Al . Ak n let DP A be the partition of n with parts k A1 1 . Ak 1 excluding any zeros notice that the parts may not be in the standard nonincreasing order . The map is more easily defined from the graphic representation of a partition known as a Ferrers diagram the first column of the diagram becomes a row with reordering as needed to write the image in nonincreasing order. See Fig. 1. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R80 1 o o o o o o oo Figure 1 An example of the map on partitions Dp 6 3 1 1 5 4 2 . Analysis of this shift operator on partitions was first published by Jprgen Brandt in 1982 4 although the author claimed that the problem had already been circulating for some time. The next year the idea was .

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