## 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

### Keywords:

broken diamond partitions; congruences; modular forms

### Citations:

Zbl 1218.05024; Zbl 1110.05010; Zbl 1206.05020

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### References:

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