A crank for bipartitions with designated summands

TL;DR

This paper introduces a new $pd$-crank for bipartitions with designated summands, establishes inequalities and positivity results, and analyzes the monotonicity and unimodality of related partition sequences.

Contribution

It defines the $pd$-crank for bipartitions with designated summands and investigates its properties, inequalities, and related moments, advancing understanding of partition combinatorics.

Findings

Inequalities for the $pd$-crank modulo 2 and 3.

Positivity of weighted $pd$-crank moments.

Unimodality of the sequence $M_{bd}(m,n)$ for most $n$.

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 $\{M_{bd}(m,n)\}_{|m|\leq n}$ is unimodal for $n\not= 1,5,7$.