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

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. | On Subsequence Sums of a Zero-sum Free Sequence Fang Sun Center for Combinatorics LPMC Nankai University Tianjin . China sunfang2005@ Submitted Jan 16 2007 Accepted Jul 18 2007 Published Jul 26 2007 Mathematics Subject Classification 11B Abstract Let G be a finite abelian group with exponent m and let S be a sequence of elements in G. Let f S denote the number of elements in G which can be expressed as the sum over a nonempty subsequence of S. In this paper we show that if SI m and S contains no nonempty subsequence with zero sum then f S 2m 1. This answers an open question formulated by Gao and Leader. They proved the same result with the restriction m 6 1. 1 Introduction Let G be a finite abelian group of order n and exponent m additively written. Let S a1 . ak be a sequence of elements in G. By 52 S we denote the set that consists of all elements of G that can be expressed as the sum over a nonempty subsequence of S . Fu . ail 1 ii . i k . We write f S I 52 S . If 0 2 52 S we call S a zero-sum free sequence. Let 52 S denote the set that consists of all elements in G which can be expressed as the sum over a subsequence of S of length n . Pn S ai1 ain 1 ii . in k . If U is a subsequence of S we write SU-i for the subsequence obtained by deleting the terms of U from S if U and V are disjoint subsequences of S we write UV for the subsequence obtained by adjoining the terms of U to V if U is a subsequence of S we write U S. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R52 1 Let D G be the Davenport s constant of G . the smallest integer d such that every sequence S of elements in G with SI d satisfies 0 2 S let s G be the smallest integer t such that every sequence of elements in G with S I t satisfies 0 2 n S . In 1961 Erdos Ginzburg and Ziv proved s G 2n 1 for any finite abelian group of order n. This result is now well known as the Erdos-Ginzburg-Ziv theorem. In 1996 Gao proved s G D G n 1 for any finite abelian group of order n. In 1999 .