A crank for bipartitions with designated summands

Andrews, Lewis, and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. A bipartition of n is an ordered pair of partitions (pi(1), pi(2)) with the sum of all of the parts being n. In this paper, we introduce a generalized crank named the pd-crank for bipartitions with designated summands and give some inequalities for the pd-crank of bipartitions with designated summands modulo 2 and 3. We also define the pd-crank moments weighted by the parity of pdcranks mu(2k,bd)(-1, n) and show the positivity of (-1)(n) mu(2k, bd)(-1, n). Let M-bd(m, n) denote the number of bipartitions of n with designated summands with pd-crank m. We prove a monotonicity property of pd-cranks of bipartitions with designated summands and find that the sequence {M-bd(m, n)}(broken vertical bar m broken vertical bar <= n) is unimodal for n not equal 1, 5, 7.