On the cardinality of subsequence sums

Abstract

Let [Formula: see text] be a finite sequence of integers, where [Formula: see text] and [Formula: see text] with [Formula: see text] for [Formula: see text]. A subsequence sum of [Formula: see text] is the sum of all terms of a nonempty subsequence of [Formula: see text]. Denoted by [Formula: see text] the set of all subsequence sums of [Formula: see text]. In this paper, for given [Formula: see text], we give the lower bound for [Formula: see text] with in terms of [Formula: see text] and the numbers of positive, negative integers in [Formula: see text]. We also determine the structure of the finite sequence [Formula: see text] of integers for which [Formula: see text] is minimal. This generalizes the results of Mistri, Pandey and Prakash. Moreover, we give a correction to a result of their paper.