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Two congruences involving Andrews-Paule’s broken 3-diamond partitions and 5-diamond partitions. (English) Zbl 1256.11055

The authors prove two congruences involving broken 3-diamond partitions and broken 5-diamond partitions that have been conjectured by P. Paule and S. Radu [Ramanujan J. 23, No. 1-3, 409–416 (2010; Zbl 1218.05024)].
Theorem 1.1 (Conjecture 3.1, loc. cit.). \[ \prod_{n=1}^\infty (1-q^n)^4(1-q^{2n})^6\equiv 6\sum_{n=0}^\infty \Delta_3(7n+5)q^n\pmod 7. \]
Theorem 1.2 (Conjecture 3.3, loc. cit.) \[ E_4(q^2)\prod_{n=1}^\infty (1-q^n)^8(1-q^{2n})^2\equiv 8\sum_{n=0}^\infty \Delta_5(11n+6)q^n\pmod{11}. \]
Methods of Lovejoy and Ono have been adapted to prove the two theorems.
This new class of combinatorial objects called broken \(k\)-diamonds was introduced by G. E. Andrews and P. Paule [Acta Arith. 126, No. 3, 281–294 (2007; Zbl 1110.05010)]. Their generating functions connect to modular forms and give rise to a variety of partition congruences. S. H. Chan [Discrete Math. 308, No. 23, 5735–5741 (2008; Zbl 1206.05020)] proved the first infinite family of congruences when \(k=2\).

MSC:

11P83 Partitions; congruences and congruential restrictions
11F11 Holomorphic modular forms of integral weight
11F33 Congruences for modular and \(p\)-adic modular forms

Software:

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References:

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[10] P. Paule and S. Radu, Infinite families of strange partition congruences for broken 2-diamonds, Ramanujan J. 23 (2010), no. 1-3, 409-416. · Zbl 1218.05024
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