On the minimum size of subset and subsequence sums in integers

Jagannath Bhanja; Ram Krishna Pandey

doi:10.5802/crmath.361

Combinatorics, Number theory

On the minimum size of subset and subsequence sums in integers

Jagannath Bhanja ¹ ; Ram Krishna Pandey ²

¹ Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj-211019, India
² Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1099-1111.

Abstract

Let $𝒜$ be a sequence of $r k$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $𝒜$ . The sum of all terms of a subsequence of $𝒜$ is called a subsequence sum of $𝒜$ . For a nonnegative integer $α \leq r k$ , let $Σ_{α} (𝒜)$ be the set of all subsequence sums of $𝒜$ that correspond to the subsequences of length $α$ or more. When $r = 1$ , we call the subsequence sums as subset sums and we write $Σ_{α} (A)$ for $Σ_{α} (𝒜)$ . In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $Σ_{α} (A)$ and $Σ_{α} (𝒜)$ . As special cases, we also obtain some already known results in this study.

Received: 2021-09-30
Revised: 2022-01-12
Accepted: 2022-03-31
Published online: 2022-10-04

DOI: 10.5802/crmath.361

Classification: 11B75, 11B13, 11B30

Author's affiliations:

Jagannath Bhanja ¹; Ram Krishna Pandey ²

¹ Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj-211019, India
² Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Jagannath Bhanja and Ram Krishna Pandey},
     title = {On the minimum size of subset and subsequence sums in integers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1099--1111},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.361},
     language = {en},
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Jagannath Bhanja; Ram Krishna Pandey. On the minimum size of subset and subsequence sums in integers. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1099-1111. doi : 10.5802/crmath.361. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.361/

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