On the minimum size of subset and subsequence sums in integers
Let $\mathcal{A}$ be a 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 nonnegative integer $\alpha \le rk$, let $\Sigma _{\al...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-10-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.361/ |
| _version_ | 1797651486198464512 |
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| author | Bhanja, Jagannath Pandey, Ram Krishna |
| author_facet | Bhanja, Jagannath Pandey, Ram Krishna |
| author_sort | Bhanja, Jagannath |
| collection | DOAJ |
| description | Let $\mathcal{A}$ be a 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 nonnegative integer $\alpha \le rk$, let $\Sigma _{\alpha } (\mathcal{A})$ 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 subsequence sums as subset sums and we write $\Sigma _{\alpha } (A)$ for $\Sigma _{\alpha } (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $\Sigma _{\alpha } (A)$ and $\Sigma _{\alpha } (\mathcal{A})$. As special cases, we also obtain some already known results in this study. |
| first_indexed | 2024-03-11T16:16:29Z |
| format | Article |
| id | doaj.art-a156bbc738204681ba8f45ab6060b9ee |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:29Z |
| publishDate | 2022-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-a156bbc738204681ba8f45ab6060b9ee2023-10-24T14:20:27ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-10-01360G101099111110.5802/crmath.36110.5802/crmath.361On the minimum size of subset and subsequence sums in integersBhanja, Jagannath0Pandey, Ram Krishna1Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj-211019, IndiaDepartment of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, IndiaLet $\mathcal{A}$ be a 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 nonnegative integer $\alpha \le rk$, let $\Sigma _{\alpha } (\mathcal{A})$ 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 subsequence sums as subset sums and we write $\Sigma _{\alpha } (A)$ for $\Sigma _{\alpha } (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $\Sigma _{\alpha } (A)$ and $\Sigma _{\alpha } (\mathcal{A})$. As special cases, we also obtain some already known results in this study.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.361/ |
| spellingShingle | Bhanja, Jagannath Pandey, Ram Krishna On the minimum size of subset and subsequence sums in integers Comptes Rendus. Mathématique |
| title | On the minimum size of subset and subsequence sums in integers |
| title_full | On the minimum size of subset and subsequence sums in integers |
| title_fullStr | On the minimum size of subset and subsequence sums in integers |
| title_full_unstemmed | On the minimum size of subset and subsequence sums in integers |
| title_short | On the minimum size of subset and subsequence sums in integers |
| title_sort | on the minimum size of subset and subsequence sums in integers |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.361/ |
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