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Divisibility of certain ℓℓ\ell -regular partitions by 2

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Ajit Singh at University of Virginia
Ajit Singh
Rupam Barman at Indian Institute of Technology Guwahati
Rupam Barman
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Abstract

For a positive integer ℓ\ell , let bℓ(n)b_{\ell }(n) denote the number of ℓ\ell -regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for b3(n)b_3(n) and b21(n)b_{21}(n). We prove a specific case of a conjecture of Keith and Zanello on self-similarities of b3(n)b_3(n) modulo 2. We next prove that the series ∑n=0∞b9(2n+1)qn\sum _{n=0}^{\infty }b_9(2n+1)q^n is lacunary modulo arbitrary powers of 2. We also prove that the series ∑n=0∞b9(4n)qn\sum _{n=0}^{\infty }b_9(4n)q^n is lacunary modulo 2.
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The Ramanujan Journal (2022) 59:813–829
https://doi.org/10.1007/s11139-022-00580-6
Divisibility of certain -regular partitions by 2
Ajit Singh1·Rupam Barman1
Received: 11 August 2021 / Accepted: 14 March 2022 / Published online: 3 May 2022
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022
Abstract
For a positive integer ,letb(n)denote the number of -regular partitions of a non-
negative integer n. Motivated by some recent conjectures of Keith and Zanello, we
establish infinite families of congruences modulo 2 for b3(n)and b21(n). We prove a
specific case of a conjecture of Keith and Zanello on self-similarities of b3(n)modulo
2. We next prove that the series ∞
n=0b9(2n+1)qnis lacunary modulo arbitrary
powers of 2. We also prove that the series ∞
n=0b9(4n)qnis lacunary modulo 2.
Keywords -Regular partitions ·Eta-quotients ·Modular forms ·Arithmetic density
Mathematics Subject Classification Primary 05A17 ·11P83 ·11F11
1 Introduction and statement of results
A partition of a positive integer nis any non-increasing sequence of positive integers
whose sum is n. The number of such partitions of nis denoted by p(n). The parti-
tion function has many congruence properties modulo primes and powers of primes.
Ramanujan discovered beautiful congruences satisfied by p(n)modulo 5,7 and 11.
Let be a fixed positive integer. An -regular partition of a positive integer nis a
partition of nsuch that none of its part is divisible by .Letb(n)be the number of
-regular partitions of n. The generating function for b(n)is given by
G(q):=
∞
n=0
b(n)qn=f
f1
,(1.1)
BAjit Singh
ajit18@iitg.ac.in
Rupam Barman
rupam@iitg.ac.in
1Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam 781039,
India
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... .}. Putting p = 29 in Theorem 1.1, we have b 21 (4(29 2 n + 29β + 66) + 1) ≡ 0 (mod 2), (1.3) for all 0 ≤ β < 29, β = 11. In [22,Theorem 1.4], the first and the third author proved the congruence (1.3) using a technique developed by Radu in [20]. In section 2, we also prove Theorem 1.1 when p = 59, 79 using Radu's technique. ...
... In [22,Theorem 1.4], the first and the third author proved Theorem 1.1 for p = 29 using the approach developed in [20,21]. We now give another proof of Theorem 1.1 for the primes p = 59, 79 using that approach. ...
Certain Diophantine equations and new parity results for 21-regular partitions
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  • Jan 2023
For a positive integer t≥2t\geq 2, let bt(n)b_{t}(n) denote the number of t-regular partitions of a nonnegative integer n. In a recent paper, Keith and Zanello investigated the parity of bt(n)b_{t}(n) when t≤28t\leq 28. They discovered new infinite families of Ramanujan type congruences modulo 2 for b21(n)b_{21}(n) involving every prime p with p≡13,17,19,23(mod24)p\equiv 13, 17, 19, 23 \pmod{24}. In this paper, we investigate the parity of b21(n)b_{21}(n) involving the primes p with p≡1,5,7,11(mod24)p\equiv 1, 5, 7, 11 \pmod{24}. We prove new infinite families of Ramanujan type congruences modulo 2 for b21(n)b_{21}(n) involving the odd primes p for which the Diophantine equation 8x2+27y2=jp8x^2+27y^2=jp has primitive solutions for some j∈{1,4,8}j\in\left\lbrace1,4,8\right\rbrace, and we also prove that the Dirichlet density of such primes is equal to 1/6. Recently, Yao provided new infinite families of congruences modulo 2 for b3(n)b_{3}(n) and those congruences involve every prime p≥5p\geq 5 based on Newman's results. Following a similar approach, we prove new infinite families of congruences modulo 2 for b21(n)b_{21}(n), and these congruences imply that b21(n)b_{21}(n) is odd infinitely often.
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... Ahmed and Baruah [1], Alladi [2], Andrews, Hirschhorn, and Sellers [6], Ballantine and Merca [8], Barman,Singh,and Singh [9], Baruah and Das [11], Calkin, Drake, James, Law, Lee, Penniston, and Radder [14], Carlson and Webb [15], Cui and Gu [16,17,18,19], Dai [20], Dai and Yan [21], Dandurand and Penniston [22], Furcy and Penniston [23], Gordon and Ono [24], Granville and Ono [25], Hirschhorn and Sellers [26], Hou, Sun, and Zhang [27], Iwata [28], Keith [30], Keith and Zanello [31,32], Lin and Wang [33], Lovejoy [34], Lovejoy and Penniston [35], Mestrige [36], Ono and Penniston [38,39], Penniston [40,41,42], Singh and Barman [44,45], Singh, Singh, and Barman [46], Wang [48], Webb [49], Xia [50], Xia and Yao [51,52], Yao [53,54], Zhao, Jin, and Yao [55]. ...
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A partition is said to be ℓ\ell-regular if none of its parts is a multiple of ℓ\ell. Let b5′(n)b^\prime_5(n) denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of n. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of b5′(n)b^\prime_5(n). We provide full characterization of the parity of b5′(2n+1)b^\prime_5(2n+1), present several congruences modulo 4, and prove that the generating function of the sequence (b5′(5n+1))(b^\prime_5(5n+1)) is lacunary modulo any arbitrary positive powers of 5.
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... In a recent paper [13], Keith and Zanello studied t-regular partitions for certain values of t ≤ 28, and made several conjectures on b t (n). In [19,20], the first and the second author have proved two conjectures of Keith and Zanello on b 3 (n) and b 25 (n) respectively, and certain specific cases related to their conjectures. Keith and Zanello proved various congruences for b 9 (n) and made the following conjecture regarding the self-similarity of b 9 (n). ...
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For a positive integer t≥2t\geq 2, let bt(n)b_{t}(n) denote the number of t-regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for b9(n)b_9(n) and b19(n)b_{19}(n). We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of b9(n)b_9(n) and b19(n)b_{19}(n) modulo 2. We also relate bt(n)b_{t}(n) to the ordinary partition function, and prove that bt(n)b_{t}(n) satisfies the Ramanujan's famous congruences for some infinite families of t. For t∈{6,10,14,15,18,20,22,26,27,28}t\in \{6,10,14,15,18,20,22,26,27,28\}, Keith and Zanello conjectured that there are no integers A>0A>0 and B≥0B\geq 0 for which bt(An+B)≡0(mod2)b_t(An+ B)\equiv 0\pmod 2 for all n≥0n\geq 0. We prove that, for any t≥2t\geq 2 and prime ℓ\ell, there are infinitely many arithmetic progressions An+B for which ∑n=0∞bt(An+B)qn≢0(modℓ)\sum_{n=0}^{\infty}b_t(An+B)q^n\not\equiv0 \pmod{\ell}. Next, we obtain quantitative estimates for the distributions of b6(n),b10(n)b_{6}(n), b_{10}(n) and b14(n)b_{14}(n) modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and 13-regular partition functions.
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