A note on the exact formulas for certain $2$-color partitions

Russelle Guadalupe

doi:10.5802/crmath.658

Article de recherche - Théorie des nombres

A note on the exact formulas for certain

2

-color partitions
[Note sur les formules exactes de certaines partitions à 2 couleurs]

Russelle Guadalupe ¹

¹ Institute of Mathematics, University of the Philippines-Diliman, Quezon City, 1101, Philippines

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1485-1490.

Résumé
Abstract

Soit $p \leq 23$ un nombre premier et $a_{p} (n)$ compte le nombre de partitions de n où les parties qui sont multiples de $p$ donnent $2$ couleurs. En utilisant un résultat de Sussman, nous dérivons la formule exacte pour $a_{p} (n)$ et obtenons une formule asymptotique pour $log a_{p} (n)$ . Nos résultats étendent partiellement le travail de Mauth, qui a prouvé la formule asymptotique pour $log a_{2} (n)$ conjecturée par Banerjee et al.

Let $p \leq 23$ be a prime and $a_{p} (n)$ count the number of partitions of $n$ where parts that are multiple of $p$ come up with $2$ colors. Using a result of Sussman, we derive the exact formula for $a_{p} (n)$ and obtain an asymptotic formula for $log a_{p} (n)$ . Our results partially extend the work of Mauth, who proved the asymptotic formula for $log a_{2} (n)$ conjectured by Banerjee et al.

Reçu le : 2024-04-02
Révisé le : 2024-05-23
Accepté le : 2024-06-17
Publié le : 2024-11-14

Zbl

DOI : 10.5802/crmath.658

Classification : 11P55, 11P82, 05A16
Keywords: Circle method,

η

-quotients, partitions, asymptotic formula
Mots-clés : Méthode des cercles,

η

-quotients, partitions, formule asymptotique

Affiliations des auteurs :

Russelle Guadalupe ¹

¹ Institute of Mathematics, University of the Philippines-Diliman, Quezon City, 1101, Philippines

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A note on the exact formulas for certain $2$-color partitions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1485--1490},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.658},
     zbl = {07945490},
     language = {en},
}

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Russelle Guadalupe. A note on the exact formulas for certain $2$-color partitions. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1485-1490. doi : 10.5802/crmath.658. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.658/

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