2019/08/31 by Jagannath Bhanja, Bhanja, Jagannath, Ram Krishna Pandey +1
Mathematics · #11B13 #11B75 #11P70 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B13 #msc:11B75 #msc:11P70
paper · pdf · doi:10.48550/arxiv.1909.00194
15 pages
arxiv created 2019/08/31 · arxiv updated 2019/09/04
Let A be a nonempty finite set of k integers. Given a subset B of A, the sum of all elements of B, denoted by s(B), is called the subset sum of B. For a nonnegative integer α (≤ k), let Σα (A):=\s(B): B ⊂ A, |B|≥ α\. Now, let A=(\underbracea1,…,a1_r1~copies, \underbracea2,…,a2_r2~copies,…, \underbraceak,…,ak_rk~copies) be a finite sequence of integers with k distinct terms, where ri≥ 1 for i=1,2,…,k. Given a subsequence B of A, the sum of all terms of B, denoted by s(B), is called the subsequence sum of B. For 0≤ α≤ ∑i=1k ri, let Σα (r,A):=\s(B): B~is a subsequence of~A~of length ≥ α\, where r=(r1,r2,…,rk). Very recently, Balandraud obtained the minimum cardinality of Σα (A) in finite fields. Motivated by Baladraud's work, we find the minimum cardinality of Σα(A) in the group of integers. We also determine the structure of the finite set A of integers for which |Σα (A)| is minimal. Furthermore, we generalize these results of subset sums to the subsequence sums Σα (r,A). As special cases of our results we obtain some already known results for the usual subset and subsequence sums.