New Congruences for Broken k-Diamond Partitions

Dazhao Tang College of Mathematics and Statistics Huxi Campus Chongqing University Chongqing—401 331 PR China

                     dazhaotang@cqu.edu.cn

Abstract Andrews and Paule introduced a new class of directed graphs, called broken kdiamond partitions. Let Δ_{k}(n) denote the number of broken k-diamond partitions of n for a fixed positive integer k. In this paper, we establish new infinite families of congruences modulo 5, 7, 25 and 49 for Δ_{k}(n) via a standard q-series technique and modular forms.

1 Introduction

In 2007, Andrews and Paule [1] introduced a new class of directed graphs, called broken k-diamond partitions. They proved that the generating function of Δ_{k}(n), the number of broken k-diamond partitions of n, is given by

∑_{n=0}^{∞} Δ_{k}(n)q^{n} = \frac{(q^{2}; q^{2})_{∞}(q^{2k+1}; q^{2k+1})_{∞}}{(q; q)_{∞}^{3}(q^{2(2k+1)}; q^{2(2k+1)})_{∞}} .

Here and in the sequel, we assume |q| < 1 and adopt the following customary notation on q-series and partitions:

(a; q)_{∞} := ∏_{n=0}^{∞}(1 − aq^{n}).

The following two congruences modulo 5 were subsequently proved by Chan [4], Radu [9], and Hirschhorn [5]:

Δ_{2}(25n + 14) ≡ 0 (mod 5),
Δ_{2}(25n + 24) ≡ 0 (mod 5).

Moreover, a number of infinite families of congruences modulo 5 satisfied by Δ_{2}(n) have been proved. See, for example, Chan [4], Hirschhorn [5], Radu [9], and Xia[12].

On the other hand, Jameson[7] and Xia[11] proved the following congruences modulo 7 enjoyed by Δ_{3}(n), which were conjectured by Paule and Radu [8]:

Δ_{3}(343n + 82) ≡ 0 (mod 7),
Δ_{3}(343n + 229) ≡ 0 (mod 7),
Δ_{3}(343n + 278) ≡ 0 (mod 7),
Δ_{3}(343n + 327) ≡ 0 (mod 7).

Quite recently, a variety of infinite families of congruences modulo 7 enjoyed by Δ_{3}(n) also have been found. See, for example, Xia [11], Yao, and Wang [13].

In this paper, we establish two infinite families of congruences modulo 5 and 25 for Δ_{k}(n) as follows:

Theorem 1. For all n ≥ 0,

Δ_{k}(25n + 24) ≡ 0 (mod 5), if k ≡ 12 (mod 25).  (1)

Theorem 2. For all n ≥ 0,

Δ_{k}(125n + 99) ≡ 0 (mod 25), if k ≡ 62 (mod 125).  (2)

Moreover, we obtain the following infinite families of congruences modulo 7 and 49 for Δ_{k}(n).

Theorem 3. For all n ≥ 0,

Δ_{k}(49n + s) ≡ 0 (mod 7), if k ≡ 24 (mod 49),  (3)

where s = 19, 33, 40, and 47.

Theorem 4. For all n ≥ 0,

Δ_{k}(343n + t) ≡ 0 (mod 49), if k ≡ 171 (mod 343),  (4)

where t = 96, 292, and 341.

2 Proofs of Theorems 1–4

2.1 The case mod 5

To prove (1), we collect some useful identities. Recall that the Ramanujan theta function f(a, b) is defined by

f(a, b) := ∑_{n=−∞}^{∞} a^{n(n+1)/2}b^{n(n−1)/2},  (5)

where |ab| < 1. The Jacobi triple product identity can be stated as

f(a, b) = (−a, −b, ab; ab)_{∞}.  (6)

One specialization of (5) is given by [2]:

ψ(q) := f(q, q^{3}) = ∑_{n=0}^{∞} q^{n(n+1)/2}.

According to (6), we have

ψ(q) = \frac{(q^{2}; q^{2})_{∞}^{2}}{(q; q)_{∞}} .

The following two identities are given by Hirschhorn [5, 6] and Berndt [3].

Lemma 5.

ψ(q) = a + qb + q^{3}c,  (7)
ψ^{2}(q^{5}) = ab + q^{5}c^{2}.  (8)

where

a = f(q^{10}, q^{15}), b = f(q^{5}, q^{20}), c = ψ(q^{25}).

Eq. (7) comes from [5, Eq. (2.1)], [3, Entry 10(i), p. 262] and Eq. (8) comes from [6, Eq. (34.1.21), p. 313].

Employing the binomial theorem, we can easily establish the following congruence, which will be frequently used without explicit mention.

Lemma 6. If p is a prime, α is a positive integer, then

(q^{α}; q^{α})_{∞}^{p} ≡ (q^{pα}; q^{pα}) (mod p),
(q; q)_{∞}^{p^{α}} ≡ (q^{p}; q^{p})^{p^{α−1}} (mod p^{α}).

Now, we are ready to state the proof of (1). Notice that k ≡ 12 (mod 25), then 2k + 1 ≡ 0 (mod 25). Let 2k + 1 = 25j, where j is a

positive integer, we get, (all the following congruences are modulo 5)

∑_{n=0}^{∞} Δ_{k}(n)q^{n} = \frac{(q^{2}; q^{2})_{∞}(q^{25j}; q^{25j})_{∞}}{(q; q)_{∞}^{3}(q^{50j}; q^{50j})_{∞}} = \frac{ψ^{3}(q)}{(q^{2}; q^{2})^{5}} \frac{(q^{25j}; q^{25j})_{∞}}{(q^{50j}; q^{50j})_{∞}}

≡ \frac{ψ^{3}(q)}{(q^{10}; q^{10})_{∞}} \frac{(q^{25j}; q^{25j})_{∞}}{(q^{50j}; q^{50j})_{∞}} .

Invoking (7), we obtain

∑_{n=0}^{∞} Δ_{k}(n)q^{n} ≡ \frac{(q^{25j}; q^{25j})_{∞}}{(q^{10}; q^{10})_{∞}(q^{50j}; q^{50j})_{∞}} × (a^{3} + 3qa^{2}b + 3q^{2}ab^{2}

+ q^{3}b^{3} + 3q^{3}a^{2}c + 6q^{4}abc + 3q^{5}b^{2}c + 3q^{6}ac^{2} + 3q^{7}bc^{2} + q^{9}c^{3}).

Extracting those terms involving the powers q^{5n+4} and combining (8), we have

∑_{n=0}^{∞} Δ_{k}(5n + 4)q^{5n} ≡ \frac{(abc + q^{5}c^{3}) (q^{25j}; q^{25j})_{∞}}{(q^{10}; q^{10})_{∞}(q^{50j}; q^{50j})_{∞}} = \frac{c(ab + q^{5}c^{2})(q^{25j}; q^{25j})_{∞}}{(q^{10}; q^{10})_{∞}(q^{50j}; q^{50j})_{∞}}

= ψ(q^{25})ψ^{2}(q^{5})\frac{(q^{25j}; q^{25j})_{∞}}{(q^{10}; q^{10})_{∞}(q^{50j}; q^{50j})_{∞}} ,

then

∑_{n=0}^{∞} Δ_{k}(5n + 4)q^{n} ≡ ψ(q^{5})ψ^{2}(q)\frac{(q^{5j}; q^{5j})_{∞}}{(q^{2}; q^{2})_{∞}(q^{10j}; q^{10j})_{∞}} = ψ(q^{5})\frac{(q^{2}; q^{2})_{∞}^{3}}{(q; q)_{∞}^{2}} \frac{(q^{5j}; q^{5j})_{∞}}{(q^{10j}; q^{10j})_{∞}}

≡ ψ(q^{5})\frac{(q^{5j}; q^{5j})_{∞}}{(q^{10j}; q^{10j})_{∞}} \frac{(q; q)_{∞}^{3}(q^{2}; q^{2})_{∞}^{3}}{(q^{5}; q^{5})_{∞}} := ∑_{n=0}^{∞} a(n)q^{n},

say. Thanks to Jacobi’s identity [2, Theorem 1.3.9, p. 14]

(q; q)_{∞}^{3} = ∑_{n=0}^{∞}(−1)^{n}(2n + 1)q^{n(n+1)/2},

it follows that

(q; q)_{∞}^{3} = J_{0} + J_{1} + J_{3},
(q^{2}; q^{2})_{∞}^{3} = J_{0}^{∗} + J_{1}^{∗} + J_{2}^{∗},  (9)

where J_{i} (resp. J_{i}^{∗}) consists of those terms in which the power of q is congruent to i modulo 5. Furthermore, we see that J_{3} ≡ 0 (mod 5) and J_{1}^{∗} ≡ 0 (mod 5), so

(q; q)_{∞}^{3} ≡ J_{0} + J_{1} (mod 5),
(q^{2}; q^{2})_{∞}^{3} ≡ J_{0}^{∗} + J_{2}^{∗} (mod 5).

Therefore,

(q; q)_{∞}^{3}(q^{2}; q^{2})_{∞}^{3} ≡ (J_{0} + J_{1}) (J_{0}^{∗} + J_{2}^{∗}) (mod 5),

which contains no terms of the form q^{5n+4}. Hence a(5n + 4) ≡ 0 (mod 5), equivalently,

Δ_{k}(25n + 24) ≡ 0 (mod 5),

as desired.

To prove the remaining congruences (2)–(4), we need to use a result of Radu and Sellers [10, Lemma 2.4]. Before introducing the result of Radu and Sellers, we will briefly interpret some definitions and notation.

For a positive integer M, let R(M) be the set of integer sequences {r : r = (r_{δ_{1}}, …, r_{δ_{k}})} indexed by the positive divisors 1 = δ_{1} < ⋯ < δ_{k} = M of M. For some r ∈ R(M), define

f_{r}(q) := ∏_{δ|M}(q^{δ}; q^{δ})_{∞}^{r_{δ}} = ∑_{n=0}^{∞} c_{r}(n)q^{n}.

Given a positive integer m. Let ℤ_{m}^{∗} be the set of all invertible elements in ℤ_{m}, and 𝕊_{m} be the set of all squares in ℤ_{m}^{∗}. We also define the set

P_{m,r}(t) := \{t′ | t′ ≡ ts + \frac{s − 1}{24} ∑_{δ|M} δr_{δ} (mod m), 0 ≤ t′ ≤ m − 1, [s]_{24m} ∈ 𝕊_{24m}\} ,

where t ∈ {0, …, m − 1} and [s]_{m} = s + mℤ.

Let Γ := SL_{2}(ℤ) and Γ_{∞} := \{(\begin{matrix}1 & h \\ 0 & 1 \end{matrix}) | h ∈ ℤ\}. For a positive integer N, we define the

congruence subgroup of level N as

Γ_{0}(N) := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ | c ≡ 0 (mod N)\} .

The index of Γ_{0}(N) in Γ is given by

[Γ : Γ_{0}(N)] = N ∏_{p|N} (1 + \frac{1}{p}) ,

where the product runs through the distinct primes dividing N.

Let κ = κ(m) = gcd(m^{2} − 1, 24) and denote Δ^{∗} by the set of tuples (m, M, N, t, r = (r_{δ})) satisfying conditions given in [10, p. 2255], we set

p_{m,r}(γ) = min_{λ∈\{0,...,m−1\}} \frac{1}{24} ∑_{δ|M} r_{δ} \frac{gcd^{2}(δ(a + κλc), mc)}{δm}

and

p_{r′}^{∗}(γ) = \frac{1}{24} ∑_{δ|N} \frac{r′_{δ} gcd^{2}(δ, c)}{δ},

where γ = (\begin{matrix}a & b \\ c & d \end{matrix}), r ∈ R(M), and r′ ∈ R(N).

The lemma of Radu and Sellers is stated as follows:

Lemma 7. Let u be a positive integer, (m, M, N, t, r = (r_{δ})) ∈ Δ^{∗}, r′ = (r′_{δ}) ∈ R(N), n be the number of double cosets in Γ_{0}(N)\Γ/Γ_{∞} and {γ_{1}, …, γ_{n}} ⊂ Γ be a complete set of representatives of the double coset Γ_{0}(N)\Γ/Γ_{∞}. Assume that p_{m,r}(γ_{i}) + p_{r′}^{∗}(γ_{i}) ≥ 0 for all i = 1, …, n. Let t_{min} := min_{t′∈P_{m,r}(t)} t′ and

v := \frac{1}{24} ((∑_{δ|M} r_{δ} + ∑_{δ|N} r′_{δ}) [Γ : Γ_{0}(N)] − ∑_{δ|N} δr′_{δ}) − \frac{1}{24m} ∑_{δ|M} δr_{δ} − \frac{t_{min}}{m} .

Then if

∑_{n=0}^{⌊v⌋} c_{r}(mn + t′)q^{n} ≡ 0 (mod u)

for all t′ ∈ P_{m,r}(t), then

∑_{n=0}^{∞} c_{r}(mn + t′)q^{n} ≡ 0 (mod u)

for all t′ ∈ P_{m,r}(t).

2.2 The cases mod 25

Let

∑_{n=0}^{∞} b(n)q^{n} = \frac{(q^{2}; q^{2})_{∞}}{(q; q)_{∞}^{3}} .

By Lemma 6, we obtain

∑_{n=0}^{∞} b(n)q^{n} = \frac{(q^{2}; q^{2})_{∞}}{(q; q)_{∞}^{3}} ≡ \frac{(q; q)_{∞}^{22}(q^{2}; q^{2})_{∞}}{(q^{5}; q^{5})_{∞}^{5}} (mod 25).  (10)

In this case, we may take

(m, M, N, t, r = (r_{1}, r_{2}, r_{5}, r_{10})) = (125, 10, 10, 99, (22, 1, −5, 0)) ∈ Δ^{∗}.

By the definition of P_{m,r}(t), we have P_{m,r}(t) = {99}. Now we can choose

r′ = (r′_{1}, r′_{2}, r′_{5}, r′_{10}) = (13, 0, 0, 0).

Let

γ_{δ} = (\begin{matrix}1 & 0 \\ δ & 1 \end{matrix}) .

Radu and Sellers [10, Lemma 2.6] also proved that {γ_{δ} : δ | N} contains a complete set of representatives of the double coset Γ_{0}(N)\Γ/Γ_{∞}.

One readily verifies that all assumptions of Lemma 7 are satisfied. Furthermore we obtain the upper bound ⌊v⌋ = 21.

For all congruences in (11), we check that they hold for n from 0 to their corresponding upper bound ⌊v⌋ via Mathematica. It follows by Lemma 7 and (10) that

b(125n + 99) ≡ 0 (mod 25)  (11)

holds for all n ≥ 0. When k ≡ 62 (mod 125), i.e., 2k + 1 ≡ 0 (mod 125), (2) is a direct consequence of (11).

2.3 The cases mod 7

Firstly, we have

∑_{n=0}^{∞} b(n)q^{n} = \frac{(q^{2}; q^{2})_{∞}}{(q; q)_{∞}^{3}} ≡ \frac{(q; q)_{∞}^{4}(q^{2}; q^{2})_{∞}}{(q^{7}; q^{7})_{∞}} (mod 7).

To prove (3), it suffices to show

b(49n + s) ≡ 0 (mod 7)  (12)

for s = 19, 33, 40, and 47.

We first show the cases s = 19, 33, and 40. Taking

(m, M, N, t, r = (r_{1}, r_{2}, r_{7}, r_{14})) = (49, 14, 14, 33, (4, 1, −1, 0)) ∈ Δ^{∗}.

We compute that P_{m,r}(t) = {19, 33, 40}. Now we can choose

r′ = (r′_{1}, r′_{2}, r′_{7}, r′_{14}) = (3, 0, 0, 0).

and taking γ as in Subsection 2.2, we verify that all these constants satisfy the assumption of Lemma 7. We thus obtain ⌊v⌋ = 6. With the help of Mathematica, we see that (12) holds up to the bound ⌊v⌋ with t ∈ {19, 33, 40}, and therefore it holds for all n ≥ 0 by Lemma 7.

Now we will turn to the case s = 47. Again we take

(m, M, N, t, r = (r_{1}, r_{2}, r_{7}, r_{14})) = (49, 14, 14, 47, (4, 1, −1, 0)) ∈ Δ^{∗}

and

r′ = (r′_{1}, r′_{2}, r′_{7}, r′_{14}) = (3, 0, 0, 0).

In this case we obtain P_{m,r}(t) = {47}. One readily computes that ⌊v⌋ = 6. Thus we verify the first 7 terms of (12) via Mathematica. It follows by Lemma 7 that it holds for all n ≥ 0.

2.4 The case mod 49

Similarly, we have

∑_{n=0}^{∞} b(n)q^{n} = \frac{(q^{2}; q^{2})_{∞}}{(q; q)_{∞}^{3}} ≡ \frac{(q; q)_{∞}^{46}(q^{2}; q^{2})_{∞}}{(q^{7}; q^{7})_{∞}^{7}} (mod 49).

To prove (4), it needs to show

b(343n + t) ≡ 0 (mod 49)  (13)

for t = 96, 292, and 341.

In these cases, we may set

(m, M, N, t, r = (r_{1}, r_{2}, r_{7}, r_{14})) = (343, 14, 14, 96, (46, 1, −7, 0)) ∈ Δ^{∗}.

We compute that P_{m,r}(t) = {96, 292, 341}. Now we can choose

r′ = (r′_{1}, r′_{2}, r′_{7}, r′_{14}) = (18, 0, 0, 0).

We thus obtain ⌊v⌋ = 56. Similarly we can verify the first 57 terms of (13) through Mathematica. Therefore,

Δ_{k}(343n + 96) ≡ Δ_{k}(343n + 292) ≡ Δ_{k}(343n + 341) ≡ 0 (mod 49)

holds for all n ≥ 0 if k ≡ 171 (mod 343).

3 Final remarks

We would like to emphasize that unlike most congruences of Δ_{k}(n) studied in the literature, where k is usually fixed, the congruences (1)–(4) are valid for infinitely many k.

Moreover, despite the universality of Radu and Sellers’ lemma, we should point out that their method is not elementary because it highly relies on modular forms. On the other hand, the proofs of (2)–(4) are routine to some extent. It would be interesting to find elementary proofs of these congruences.

4 Acknowledgements

The author would like to thank Shishuo Fu, Shane Chern and Michael D. Hirschhorn for their helpful comments and suggestions on the original manuscript. The author would like to acknowledge the referee for his/her careful reading and helpful comments on an earlier version of the paper. This work was supported by the National Natural Science Foundation of China (No. 11501061).

References

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[2] B. C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, 2006.

[3] B. C. Berndt, Ramanujan Notebooks Part III, Springer–Verlag, 1991.

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[5] M. D. Hirschhorn, Broken 2-diamond partitions modulo 5, Ramanujan J. 45 (2018), 517–520.

[6] M. D. Hirschhorn, The Power of q, Developments in Mathematics, Vol. 49, Springer, 2017.

[7] M. Jameson, Congruences for broken k-diamond partitions, Ann. Comb. 17 (2013), 333–338.

[8] P. Paule and C. S. Radu, Infinite families of strange partition congruences for broken 2-diamonds, Ramanujan. J 23 (2010), 409–416.

[9] C. S. Radu, An algorithmic approach to Ramanujan-Kolberg identities, J. Symb. Comput. 68 (2015), 225–253.

[10] C. S. Radu and J. A. Sellers, Congruence properties modulo 5 and 7 for the pod function, Int. J. Number Theory 7 (2011), 2249–2259.

[11] E. X. W. Xia, Infinite families of congruences modulo 7 for broken 3-diamond partitions, Ramanujan J. 40 (2016), 389–403.

[12] E. X. W. Xia, More infinite families of congruences modulo 5 for broken 2-diamond partitions, J. Number Theory 170 (2017), 250–262.

[13] O. X. M. Yao and Y. J. Wang, Newman’s identity and infinite families of congruences modulo 7 for broken 3-diamond partitions, Ramanujan J. 43 (2017), 619–631.

2010 Mathematics Subject Classification: Primary 05A17; Secondary 11P83. Keywords: partition, broken k-diamond partition, congruences.

Received September 11 2017; revised versions received June 26 2018; June 29 2018. Published in Journal of Integer Sequences, July 1 2018.

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