Đang chuẩn bị liên kết để tải về tài liệu:
Báo cáo toán học: "On Subsequence Sums of a Zero-sum Free Sequence II"
Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On Subsequence Sums of a Zero-sum Free Sequence II. | On Subsequence Sums of a Zero-sum Free Sequence II Weidong Gao1 Yuanlin Li2 Jiangtao Peng3 and Fang Sun4 1 3 4Center for Combinatorics LPMC Nankai University Tianjin P.R. China 2Department of Mathematics Brock University St. Catharines Ontario Canada L2S 3A1 1gao@cfc.nankai.edu.cn 2yli@brocku.ca 3pjt821111@cfc.nankai.edu.cn 4sunfang2005@163.com Submitted Apr 29 2008 Accepted Sep 2 2008 Published Sep 15 2008 Mathematics Subject Classification 11B Abstract Let G be an additive finite abelian group with exponent exp G n. For a sequence S over G let f S denote the number of non-zero group elements which can be expressed as a sum of a nontrivial subsequence of S. We show that for every zero-sum free sequence S over G of length SI n 1 we have f S 3n 1. 1 Introduction and Main results Let G be an additive finite abelian group with exponent exp G n and let S be a sequence over G we follow the conventions of 5 concerning sequences over abelian groups details are recalled in Section 2 . We denote by E S the set of all subsums of S and by f G S f S the number of nonzero group elements which can be expressed as a sum of a nontrivial subsequence of S thus f S E S n 0 . In 1972 R.B. Eggleton and P. Erdos see 2 first tackled the problem of determining the minimal cardinality of E S for squarefree zero-sum free sequences that is for zerosum free subsets of G see 7 for recent progress. For general sequences the problem was first studied by J.E. Olson and E.T. White in 1977 see Lemma 2.5 . In a recent new approach 16 the fourth author of this paper proved that every zero-sum free sequence S over G of length S n satisfies f S 2n 1. A main result of the present paper runs as follows. Theorem 1.1. Let G Cni . Cnr be a finite abelian group with 1 n1 . nr. If r 2 and nr-_1 3 then every zero-sum free sequence S over G of length S nr 1 satisfies f S 3nr 1. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R117 1 This partly confirms a former conjecture of B. Bollobás and I. Leader which is .