THE UNIMODALITY OF THE r 3-CRANK OF 3-REGULAR OVERPARTITIONS

An l - regular overpartition of n is an overpartition of n with no parts divisible by l. Recently, the authors introduced a partition statistic called r l -crank of l - regular overpartitions. Let M r l ( m, n ) denote the number of l - regular overpartitions of n with r l -crank m. In this paper, we investigate the monotonicity property and the unimodality of M r 3 ( m, n ). We prove that M r 3 ( m, n ) >= M r 3 ( m, n - 1) for any integers m and n >= 6 and the sequence { M r 3 ( m, n ) } | m |<= n is unimodal for all n >= 14.