On the cardinality of subsequence sums II

Xing-Wang Jiang; Ya-Li Li

doi:10.5802/crmath.613

Article de recherche - Théorie des nombres

On the cardinality of subsequence sums II
[Sur le cardinal de sommes de sous-suites II]

Xing-Wang Jiang ¹ ; Ya-Li Li ²

¹ Department of Mathematics, Luoyang Normal University, Luoyang 471934, P. R. China
² School of Mathematics and Statistics, Henan University, Kaifeng 475001, P. R. China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1279-1285.

Résumé
Abstract

Soit $A = ({\underset{︸}{a_{1}, ..., a_{1}}}_{r_{1}}, {\underset{︸}{a_{2}, ..., a_{2}}}_{r_{2}}, ..., {\underset{︸}{a_{k}, ..., a_{k}}}_{r_{k}})$ ) une suite finie d’entiers avec $a_{1} < a_{2} < \dots < a_{k}$ et $r_{i} \geq 1 (1 \leq i \leq k)$ . La somme de tous les termes d’une sous-suite $B$ de $A$ est appelée somme de sous-suite de $A$ et nous la désignons par $σ (B)$ . Pour $0 \leq α \leq \sum_{i = 1}^{k} r_{i}$ , soit $Σ_{α} (A) = {σ (B) | B une sous-suite de A de longueur \geq α}$ . Dans cet article, nous résolvons complètement les deux problèmes posés par Bhanja et Pandey concernant la recherche de la borne inférieure de $| Σ_{α} (A) |$ et la détermination de la structure de la suite $A$ pour laquelle la borne inférieure de $| Σ_{α} (A) |$ est atteinte.

Let $A = ({\underset{︸}{a_{1}, ..., a_{1}}}_{r_{1}}, {\underset{︸}{a_{2}, ..., a_{2}}}_{r_{2}}, ..., {\underset{︸}{a_{k}, ..., a_{k}}}_{r_{k}})$ be a finite sequence of integers with $a_{1} < a_{2} < \dots < a_{k}$ and $r_{i} \geq 1 (1 \leq i \leq k)$ . The sum of all terms of a subsequence $B$ of $A$ is called a subsequence sum of $A$ and we denote it by $σ (B)$ . For $0 \leq α \leq \sum_{i = 1}^{k} r_{i}$ , let $Σ_{α} (A) = {σ (B) | B is a subsequence of A of length \geq α}$ . In this paper, we completely settle the two problems posed by Bhanja and Pandey about finding the optimal lower bound of $| Σ_{α} (A) |$ and determining the structure of the sequence $A$ for which the lower bound of $| Σ_{α} (A) |$ is optimal.

Reçu le : 2023-07-05
Révisé le : 2023-10-22
Accepté le : 2024-01-15
Publié le : 2024-11-14

DOI : 10.5802/crmath.613

Classification : 11P70, 11B25
Keywords: Subsequence sums, direct problem, inverse problem
Mot clés : Sommes de sous-suites, problème direct, problème inverse

Affiliations des auteurs :

Xing-Wang Jiang ¹ ; Ya-Li Li ²

¹ Department of Mathematics, Luoyang Normal University, Luoyang 471934, P. R. China
² School of Mathematics and Statistics, Henan University, Kaifeng 475001, P. R. China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Xing-Wang Jiang and Ya-Li Li},
     title = {On the cardinality of subsequence sums {II}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1279--1285},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.613},
     language = {en},
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Xing-Wang Jiang; Ya-Li Li. On the cardinality of subsequence sums II. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1279-1285. doi : 10.5802/crmath.613. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.613/

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