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Báo cáo toán học: "A note on major sequences and external activity in trees"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A note on major sequences and external activity in trees. | A note on major sequences and external activity in trees Janet S. Beissinger Institute for Mathematics and Science Education The University of Illinois at Chicago M C 250 950 S. Halsted Street Chicago IL 60607-7019 Beissing@uic.edu and Uri N. Peled Dept. of Mathematics Statistics and Computer Science The University of Illinois at Chicago M C 249 851 S. Morgan Street Chicago IL 60607-7045 uripeled@uic.edu Submitted September 20 1996 Accepted October 31 1996. Abstract A bijection is given from major sequences of length n a variant of parking functions to trees on 0 . ng that maps a sequence with sum 1 k to a tree with external activity k. Key Words Major sequence external activity parking function bijection Mathematical Reviews Subject Numbers Primary 05A19 Secondary 05A15 05C05 05C30 THE ELECTRONIC .JOURNAL OF COMBINATORICS 4 no. 2 1997 R4 2 We present a bijection from major seqeunces a variant of parking funtions of length n to trees on 0 . ng that takes area to external activity. Our main tool is a decomposition of major sequences due to Kreweras 6 . An integer sequence S s1 . sn is called a major sequence of length n 6 if its non-decreasing rearrangement z1 . zn satisfies zi i for all 1 i n and zn n. Another way to view z1 . zn is as a lattice path from 0 0 to n n that never drops below the main diagonal. In Figure 1 the top lattice path represents the nondecreasing rearrangement of the major sequence 7 8 8 3 3 5 3 7 and the bottom represents the identity 1 2 3 4 5 6 7 8 . 8 7 6 5 4 3 2 1 0 012345678 Figure 1 The nondecreasing rearrangement of the major sequence 7 8 8 3 3 5 3 7 with length 8 and area 8. We note that s1 . sn is a major sequence iff n s1 . n sn is a parking function as defined in Stanley 7 8 i.e. a sequence of n integers between 0 and n 1 at most i of which are n i for all 1 i n. The area of the major sequence S s1 . sn is defined as X _ n 1 a S 2_ si 2 . i 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 4 no. 2 1997 R4 3 It is non-negative and is the .