A crank for bipartitions with designated summands

Abstract

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 $pd$-cranks ${\mu }_{2k,bd}(-1,n)$ 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 $\{M_{bd}(m,n){\}}_{|m|\le n}$ is unimodal for $n\ne 1,5,7$.

17 pages