tailieunhanh - Báo cáo toán học: "On Subsequence Sums of a Zero-sum Free Sequence II"

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 . China 2Department of Mathematics Brock University St. Catharines Ontario Canada L2S 3A1 1gao@ 2yli@ 3pjt821111@ 4sunfang2005@ 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 . 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 . Olson and . White in 1977 see Lemma . 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 . 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 .