Arithmetic properties of overpartition triples

作者：Wang Liu Quan^*

来源：Acta Mathematica Sinica-English Series, 2017, 33(1): 37-50.

DOI：10.1007/s10114-016-5673-2

• Partitions
• overpartition triples
• congruences
• theta functions

Let (p) over bar (3)(n) be the number of overpartition triples of n. By elementary series manipulations, we establish some congruences for (p) over bar (3)(n) modulo small powers of 2, such as
(p) over bar (3) (16n + 14) equivalent to 0 (mod 32), (p) over bar (3) (8n + 7) equivalent to 0 (mod 64).
We also find many arithmetic properties for (p) over bar (3)(n) modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers alpha >= 1 and n >= 0, we have
(p) over bar (3) (3(2 alpha+1) (3n + 2)) equivalent to 0 (mod 9 center dot 2(4)), (p) over bar (3)(4(alpha-1) (56n + 49)) equivalent to 0 (mod 7),
(p) over bar (3) (7(2 alpha+1) (7n + 3)) equivalent to (p) over bar (3) (7(2 alpha+1) (7n + 5)) equivalent to (p) over bar (3) (7(2 alpha+1) (7n + 6)) equivalent to 0 (mod 7),
and for r is an element of{1,2,3,4,5,6},
(p) over bar (3) (11 center dot 7(4 alpha-1) (7n + r)) equivalent to 0 (mod 11).

• 出版日期2017-1