带指定被加数的二分划的曲柄
A crank for bipartitions with designated summands
摘要：
Andrews、Lewis和Lovejoy引入了配分函数$PD(n)$，作为指定被加数的$n$的分区数。$n$的二分区是一对有序的分区$(π_1, π_2)$，所有部分的总和为$n$。本文对指定和数的二分划引入了一个广义曲柄，称为$pd$ -曲柄，并给出了指定和数模2和模3的二分划的$pd$ -曲柄的一些不等式.我们还定义了$pd$ -曲柄力矩由$pd$ -曲柄$μ_{2k,bd}(-1,n)$的奇偶性加权，并证明了$(-1)^nμ_{2k,bd}(-1,n)$的正性.令$M_{bd}(m,n)$表示$n$的具有指定被加数的二分划的数目，其中$pd$ -曲柄$m$。证明了具有指定和数的二分划的$pd$ -曲柄的单调性，并发现序列$\{M_{bd}(m,n)\}_{|m|\leq n}$对于$n\not= 1,5,7$是单峰的.
英文摘要：
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 $(π_1, π_2)$ 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 $μ_{2k,bd}(-1,n)$ and show the positivity of $(-1)^nμ_{2k,bd}(-1,n)$. 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 $n\not= 1,5,7$.
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