Congruences modulo $4$ for the number of $3$-regular partitions

Cristina Ballantine; Mircea Merca

doi:10.5802/crmath.512

Combinatoire, Théorie des nombres

Congruences modulo

4

for the number of

3

-regular partitions

Cristina Ballantine ¹ ; Mircea Merca ^2,³

¹ Department of Mathematics and Computer Science, College of The Holy Cross, Worcester, MA 01610, USA
² Department of Mathematical Methods and Models, Fundamental Sciences Applied in Engineering Research Center, University Politehnica of Bucharest, RO-060042 Bucharest, Romania
³ Academy of Romanian Scientists, RO-050044, Bucharest, Romania

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1577-1583.

Résumé

The last decade has seen an abundance of congruences for $b_{ℓ} (n)$ , the number of $ℓ$ -regular partitions of $n$ . Notably absent are congruences modulo $4$ for $b_{3} (n)$ . In this paper, we introduce Ramanujan type congruences modulo $4$ for $b_{3} (2 n)$ involving some primes $p$ congruent to $11, 13, 17, 19, 23$ modulo $24$ .

Reçu le : 2023-03-14
Révisé le : 2023-04-19
Accepté le : 2023-05-06
Publié le : 2023-11-10

DOI : 10.5802/crmath.512

Classification : 11P83, 05A17, 11F33
Mots clés : partitions, regular partitions, congruences

Affiliations des auteurs :

Cristina Ballantine ¹ ; Mircea Merca ^{2, 3}

¹ Department of Mathematics and Computer Science, College of The Holy Cross, Worcester, MA 01610, USA
² Department of Mathematical Methods and Models, Fundamental Sciences Applied in Engineering Research Center, University Politehnica of Bucharest, RO-060042 Bucharest, Romania
³ Academy of Romanian Scientists, RO-050044, Bucharest, Romania

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G9_1577_0,
     author = {Cristina Ballantine and Mircea Merca},
     title = {Congruences modulo $4$ for the number of $3$-regular partitions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1577--1583},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.512},
     language = {en},
}

TY  - JOUR
AU  - Cristina Ballantine
AU  - Mircea Merca
TI  - Congruences modulo $4$ for the number of $3$-regular partitions
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1577
EP  - 1583
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.512
LA  - en
ID  - CRMATH_2023__361_G9_1577_0
ER  -

%0 Journal Article
%A Cristina Ballantine
%A Mircea Merca
%T Congruences modulo $4$ for the number of $3$-regular partitions
%J Comptes Rendus. Mathématique
%D 2023
%P 1577-1583
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.512
%G en
%F CRMATH_2023__361_G9_1577_0

Cristina Ballantine; Mircea Merca. Congruences modulo $4$ for the number of $3$-regular partitions. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1577-1583. doi : 10.5802/crmath.512. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.512/

Bibliographie
Cité par

[1] George E. Andrews The Theory of Partitions, Cambridge Mathematical Library, Cambridge University Press, 1998 | Zbl

[2] Cristina Ballantine; Mircea Merca; Cristian-Silviu Radu Parity of $3$ -regular partition numbers and Diophantine equations (2022) | arXiv

[3] Rowland Carlson; John J. Webb Infinite families of infinite families of congruences for $k$ -regular partitions, Ramanujan J., Volume 33 (2014) no. 3, pp. 329-337 | DOI | MR | Zbl

[4] Su-Ping Cui; Nancy S. S. Gu Arithmetic properties of $ℓ$ -regular partitions, Adv. Appl. Math., Volume 51 (2013) no. 4, pp. 507-523 | MR | Zbl

[5] Brian Dandurand; David Penniston $l$ -Divisibility of $l$ -regular partition functions, Ramanujan J., Volume 19 (2009) no. 1, pp. 63-70 | DOI | MR | Zbl

[6] David Furcy; David Penniston Congruences for $ℓ$ -regular partition functions modulo $3$ , Ramanujan J., Volume 27 (2012) no. 1, pp. 101-108 | DOI | MR | Zbl

[7] Michael D. Hirschhorn; James A. Sellers Elementary proofs of parity results for $5$ -regular partitions, Bull. Aust. Math. Soc., Volume 81 (2010) no. 1, pp. 58-63 | DOI | MR | Zbl

[8] William J. Keith Congruences for $m$ -regular partitions modulo $4$ , Integers, Volume 15A (2015), A11, 12 pages | MR | Zbl

[9] William J. Keith; Fabrizio Zanello Parity of the coefficients of certain eta-quotients, J. Number Theory, Volume 235 (2022), pp. 275-304 | DOI | MR | Zbl

[10] Jeremy Lovejoy; David Penniston $3$ -regular partitions and a modular $K 3$ surface, $q$ -series with applications to combinatorics, number theory, and physics (Contemporary Mathematics), Volume 291, American Mathematical Society, 2001, pp. 177-182 | MR | Zbl

[11] David Penniston The $p^{a}$ -regular partition function modulo $p^{j}$ , J. Number Theory, Volume 94 (2002) no. 2, pp. 320-325 | DOI | MR | Zbl

[12] David Penniston Arithmetic of $ℓ$ -regular partition functions, Int. J. Number Theory, Volume 4 (2008) no. 2, pp. 295-302 | DOI | MR | Zbl

[13] Silviu Radu An algorithmic approach to Ramanujan’s congruences, Ramanujan J., Volume 20 (2009) no. 2, pp. 215-251 | DOI | MR | Zbl

[14] Silviu Radu; James A. Sellers Congruence properties modulo 5 and 7 for the pod function, Int. J. Number Theory, Volume 7 (2011) no. 8, pp. 2249-2259 | DOI | MR | Zbl

[15] Øystein Rødseth Congruence properties of the partition functions $q (n)$ and $q_{0} (n)$ , Årbok Univ. Bergen, Mat.-Naturv. Ser., Volume 1969 no. 13, 27 pages | MR | Zbl

[16] Liuquan Wang Arithmetic properties of $(k, ℓ)$ -regular bipartitions, Bull. Aust. Math. Soc., Volume 95 (2017) no. 3, pp. 353-364 | DOI | MR | Zbl

[17] Liuquan Wang Congruences for $5$ -regular partitions modulo powers of $5$ , Ramanujan J., Volume 44 (2017) no. 2, pp. 343-358 | DOI | MR | Zbl

[18] Liuquan Wang Arithmetic properties of $7$ -regular partitions, Ramanujan J., Volume 47 (2018) no. 1, pp. 99-115 | DOI | MR | Zbl

[19] John J. Webb Arithmetic of the $13$ -regular partition function modulo $3$ , Ramanujan J., Volume 25 (2011), pp. 49-56 | DOI | MR | Zbl

[20] Ernest X. W. Xia Congruences for some $l$ -regular partitions modulo $l$ , J. Number Theory, Volume 152 (2015), pp. 105-117 | MR | Zbl

[21] Ernest X. W. Xia; Olivia X. M. Yao New Ramanujan-like congruences modulo powers of $2$ and $3$ for overpartitions, J. Number Theory, Volume 133 (2013) no. 6, pp. 1932-1949 | MR | Zbl

[22] Ernest X. W. Xia; Olivia X. M. Yao Parity results for $9$ -regular partitions, Ramanujan J., Volume 34 (2014) no. 1, pp. 109-117 | MR | Zbl

[23] Olivia X. M. Yao New parity results for $3$ -regular partitions, Quaest. Math., Volume 46 (2023) no. 3, pp. 465-471 | MR | Zbl

Cité par Sources :

Commentaires - Politique