EQUALITIES FOR THE r3-CRANK OF 3-REGULAR OVERPARTITIONS

Lovejoy introduced the partition function Al(n) as the number of l-regular overpartitions of n. Andrews defined (k, i)singular overpartitions counted by the partition function Ck,i(n), and pointed out that C3,1(n) = A3(n). Meanwhile, Andrews derived an interesting divisibility property that C3,1(9n+ 3) = C3,1(9n+ 6) = 0 (mod 3). Recently, we constructed the partition statistic rl-crank of l-regular overpartitions and give combinatorial interpretations for some congruences of Al(n) as well as the congruences of Andrews. In this paper, we aim to prove some equalities for the r3-crank of 3-regular overpartitions.