Arithmetic properties for $$\ell $$-regular partition functions with distinct even parts

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Rinchin Drema ◽  
Nipen Saikia
Keyword(s):  
Partition Functions ◽  
Regular Partition ◽  
Arithmetic Properties

Newman's identities, lucas sequences and congruences for certain partition functions

2020 ◽  
Vol 63 (3) ◽  
pp. 709-736
Author(s):  
Ernest X.W. Xia
Keyword(s):  
Partition Functions ◽  
Regular Partition ◽  
Lucas Sequences ◽  
Image Position ◽  
Smallest Parts Function

AbstractLet r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0, \[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \] and for k ≥ 0, \[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \] where spt(n) denotes Andrews's smallest parts function.

THE 7-REGULAR AND 13-REGULAR PARTITION FUNCTIONS MODULO 3

2016 ◽  
Vol 93 (3) ◽  
pp. 410-419 ◽  
Author(s):  
ERIC BOLL ◽  
DAVID PENNISTON
Keyword(s):  
Partition Functions ◽  
Regular Partition ◽  
Regular Partitions

Let $b_{\ell }(n)$ denote the number of $\ell$-regular partitions of $n$. In this paper we establish a formula for $b_{13}(3n+1)$ modulo $3$ and use this to find exact criteria for the $3$-divisibility of $b_{13}(3n+1)$ and $b_{13}(3n)$. We also give analogous criteria for $b_{7}(3n)$ and $b_{7}(3n+2)$.

scholarly journals Quadratic Forms and Four Partition Functions Modulo 3

Integers ◽  
2011 ◽  
Vol 11 (1) ◽  
Author(s):  
Jeremy Lovejoy ◽  
Robert Osburn
Keyword(s):  
Quadratic Forms ◽  
Partition Functions ◽  
Arithmetic Properties

AbstractRecently, Andrews, Hirschhorn and Sellers have proven congruences modulo 3 for four types of partitions using elementary series manipulations. In this paper, we generalize their congruences using arithmetic properties of certain quadratic forms.

Congruences for ℓ-regular partition functions modulo 3

2011 ◽  
Vol 27 (1) ◽  
pp. 101-108 ◽  
Author(s):  
David Furcy ◽  
David Penniston
Keyword(s):  
Partition Functions ◽  
Regular Partition

scholarly journals Arithmetic properties for 7-regular partition triples

2020 ◽  
Vol 51 (2) ◽  
pp. 717-733
Author(s):  
Shane Chern ◽  
Dazhao Tang ◽  
Ernest X. W. Xia
Keyword(s):  
Regular Partition ◽  
Arithmetic Properties ◽  
Partition Triples

scholarly journals CONGRUENCES FOR k DOTS BRACELET PARTITION FUNCTIONS

2013 ◽  
Vol 09 (08) ◽  
pp. 1885-1894 ◽  
Author(s):  
SU-PING CUI ◽  
NANCY SHAN SHAN GU
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Partition Functions ◽  
Arithmetic Properties ◽  
Partition Analysis ◽  
Congruence Relations ◽  
Modulo P ◽  
Regular Partitions

Andrews and Paule introduced broken k-diamond partitions by using MacMahon's partition analysis. Recently, Fu found a generalization which he called k dots bracelet partitions and investigated some congruences for this kind of partitions. In this paper, by finding congruence relations between the generating function for 5 dots bracelet partitions and that for 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a given prime p, we study arithmetic properties modulo p of k dots bracelet partitions.

scholarly journals On the distribution of odd values of2a-regular partition functions

2014 ◽  
Vol 143 ◽  
pp. 14-23 ◽  
Author(s):  
Haobo Dai ◽  
Chunlei Liu ◽  
Haode Yan
Keyword(s):  
Partition Functions ◽  
Regular Partition

ARITHMETIC OF ℓ-REGULAR PARTITION FUNCTIONS

2008 ◽  
Vol 04 (02) ◽  
pp. 295-302 ◽  
Author(s):  
DAVID PENNISTON
Keyword(s):  
Partition Functions ◽  
Regular Partition ◽  
Regular Partitions

Let bℓ(n) denote the number of ℓ-regular partitions of n, where ℓ is prime and 3 ≤ ℓ ≤ 23. In this paper we prove results on the distribution of bℓ(n) modulo m for any odd integer m > 1 with 3 ∤ m if ℓ ≠ 3.

scholarly journals ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS

2009 ◽  
Vol 81 (1) ◽  
pp. 58-63 ◽  
Author(s):  
MICHAEL D. HIRSCHHORN ◽  
JAMES A. SELLERS
Keyword(s):  
Modular Forms ◽  
Triple Product ◽  
Partition Functions ◽  
Regular Partition ◽  
Jacobi’S Triple Product Identity ◽  
Striking Result ◽  
Product Identity ◽  
Divisibility Properties ◽  
Regular Partitions ◽  
Positive Integers

AbstractIn a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.

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