论文标题
加权数量和分区的迹象行为
Signs behaviour of sums of weighted numbers of partitions
论文作者
论文摘要
让$ a $是积极整数的子集。按$ n $的$ a $分区，我们了解$ n $的表示形式，作为$ a $ a $的元素的总和。对于给定的$ i，n \ in \ n $，由$ c_ {a}（i，n）$，我们表示$ a $ n $的$ a $ n $的数量，恰好是$ i $零件。在论文中，我们获得了几个有关序列$ s_ {a，k}（n）= \ sum_ {i = 0}^{n}（ - 1）^{i} i^{k} c_ {a} c_ {a}（a}（i，n）$，其中$ k \ in \ n $ nesive。特别是，我们证明，对于$ \ n _ {+} $的子集的广泛类$ \ cal {a} $，我们都有每个$ a \ in \ cal {a} $我们都有$（-1）^{n} s_ {n} s_ {a，k}（a，k}（a，k}（n）\ geq 0 $ for $ n $ n $ n，k n $ n，k n $ n $ n，k.
Let $A$ be a subset of positive integers. By $A$-partition of $n$ we understand the representation of $n$ as a sum of elements from the set $A$. For given $i, n\in\N$, by $c_{A}(i,n)$ we denote the number of $A$-partitions of $n$ with exactly $i$ parts. In the paper we obtain several result concerning sign behaviour of the sequence $S_{A,k}(n)=\sum_{i=0}^{n}(-1)^{i}i^{k}c_{A}(i,n)$, where $k\in\N$ is fixed. In particular, we prove that for a broad class $\cal{A}$ of subsets of $\N_{+}$ we have that for each $A\in \cal{A}$ we have $(-1)^{n}S_{A,k}(n)\geq 0$ for each $n, k\in\N$.
