On the minimum size of subset and subsequence sums in integers

Jagannath Bhanja, Ram Krishna Pandey

Published 2021-08-16Version 1

Let $\mathcal{A}$ be a finite sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a fixed nonnegative integer $\alpha \leq rk$, let $\Sigma_{\alpha} (r,\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $\alpha$ or more. When $r=1$, we call the set of subsequence sums as the set of subset sums and we denote it by $\Sigma_{\alpha} (A)$ instead of $\Sigma_{\alpha} (1,\mathcal{A})$. In this article, we give optimal lower bounds for the sizes of $\Sigma_{\alpha} (A)$ and $\Sigma_{\alpha} (r,\mathcal{A})$. As special cases, we also obtain some already known results in this study.