On the minimum size of subset and subsequence sums in integers

Bhanja, Jagannath; Pandey, Ram Krishna

doi:10.5802/crmath.361

Combinatoire, Théorie des nombres

On the minimum size of subset and subsequence sums in integers

Bhanja, Jagannath ¹ ; Pandey, Ram Krishna 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, Tome 360 (2022) no. G10, pp. 1099-1111.

Résumé

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.

Reçu le : 2021-09-30
Révisé le : 2022-01-12
Accepté le : 2022-03-31
Publié le : 2022-10-04

DOI : 10.5802/crmath.361

Classification : 11B75, 11B13, 11B30

Affiliations des auteurs :

Bhanja, Jagannath ¹ ; Pandey, Ram Krishna ²

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Bhanja, Jagannath; Pandey, Ram Krishna. On the minimum size of subset and subsequence sums in integers. Comptes Rendus. Mathématique, Tome 360 (2022) no. G10, pp. 1099-1111. doi : 10.5802/crmath.361. http://www.numdam.org/articles/10.5802/crmath.361/

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