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

DOI : 10.5802/crmath.658

Classification : 11P55, 11P82, 05A16
Keywords: Circle method, $\eta $-quotients, partitions, asymptotic formula
Mot clés : Méthode des cercles, $\eta $-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

@article{CRMATH_2024__362_G11_1485_0,
     author = {Russelle Guadalupe},
     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},
     language = {en},
}

TY  - JOUR
AU  - Russelle Guadalupe
TI  - A note on the exact formulas for certain $2$-color partitions
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 1485
EP  - 1490
VL  - 362
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.658
LA  - en
ID  - CRMATH_2024__362_G11_1485_0
ER  -

%0 Journal Article
%A Russelle Guadalupe
%T A note on the exact formulas for certain $2$-color partitions
%J Comptes Rendus. Mathématique
%D 2024
%P 1485-1490
%V 362
%I Académie des sciences, Paris
%R 10.5802/crmath.658
%G en
%F CRMATH_2024__362_G11_1485_0

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/

Bibliographie
Cité par

[1] Zakir Ahmed; Nayandeep Deka Baruah; Manosij Ghosh Dastidar New congruences modulo 5 for the number of 2-color partitions, J. Number Theory, Volume 157 (2015), pp. 184-198 | DOI | MR | Zbl

[2] Koustav Banerjee Inequalities for the modified Bessel function of first kind of non-negative order, J. Math. Anal. Appl., Volume 524 (2023) no. 1, 127082, 28 pages | DOI | MR | Zbl

[3] Kathrin Bringmann; Ken Ono Coefficients of harmonic Maass forms, Partitions, $q$ -series, and modular forms (Developments in Mathematics), Volume 23, Springer, 2012, pp. 23-38 | DOI | MR | Zbl

[4] Koustav Banerjee; Peter Paule; Cristian-Silviu Radu; WenHuan Zeng New inequalities for $p (n)$ and $log p (n)$ , Ramanujan J., Volume 61 (2023) no. 4, pp. 1295-1338 | DOI | MR | Zbl

[5] O-Yeat Chan Some asymptotics for cranks, Acta Arith., Volume 120 (2005) no. 2, pp. 107-143 | DOI | MR | Zbl

[6] Shane Chern Asymptotics for the Fourier coefficients of eta-quotients, J. Number Theory, Volume 199 (2019), pp. 168-191 | DOI | MR | Zbl

[7] William Y. C. Chen; Larry X. W. Wang; Gary Y. B. Xie Finite differences of the logarithm of the partition function, Math. Comput., Volume 85 (2016) no. 298, pp. 825-847 | DOI | MR | Zbl

[8] Stephen DeSalvo; Igor Pak Log-concavity of the partition function, Ramanujan J., Volume 38 (2015) no. 1, pp. 61-73 | DOI | MR | Zbl

[9] G. H. Hardy; S. Ramanujan Asymptotic formulae in combinatory analysis., Proc. Lond. Math. Soc., Volume 17 (1918), pp. 75-115 | DOI

[10] B. Kim A crank analog on a certain kind of partition function arising from the cubic continued fraction (2008) (preprint, pp. 13)

[11] V. Kotesovec A method of finding the asymptotics of q-series based on the convolution of generating functions (2015) (https://arxiv.org/abs/1509.08708)

[12] Lukas Mauth Exact formula for cubic partitions, Ramanujan J., Volume 64 (2024) no. 4, pp. 1323-1334 | DOI | MR | Zbl

[13] Ken Ono The web of modularity: arithmetic of the coefficients of modular forms and $q$ -series, CBMS Regional Conference Series in Mathematics, 102, Published for the Conference Board of the Mathematical Sciences, by the American Mathematical Society, 2004, viii+216 pages | MR | Zbl

[14] Hans Rademacher On the partition function $p (n)$ , Proc. Lond. Math. Soc., Volume 43 (1937), pp. 241-254 | DOI | Zbl

[15] E. Sussman Rademacher Series for $η$ -quotients (2017) (https://arxiv.org/abs/1710.03415)

[16] Herbert S. Zuckerman On the coefficients of certain modular forms belonging to subgroups of the modular group, Trans. Am. Math. Soc., Volume 45 (1939) no. 2, pp. 298-321 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique