论文标题
通过加权词对施密特型分区进行系统研究
Systematic study of Schmidt-type partitions via weighted words
论文作者
论文摘要
令$ s =（s_n）_ {n \ geq 1} $为一个序列的序列，具有元素的元素，中的元素（\ Mathcal {m}，+，0）$。在本文中，我们为$$ \ sum _ {\ la} c（\ la）q^{\ sum_ {分配给$ \ la $的零件的``颜色''，$ q^s $是$ q $的正式功率，对于$ s \ in m $。该公式不仅使我们可以检索几个已知的Schmidt型定理，还可以为非周期序列$ s $提供新的Schmidt-type定理。例如，当$（m，+，0）=（\ mathbb {z} _ {\ geq 0}，+，0）$，$ s_n = 1 $，如果存在$ i \ geq 1 $ s sise $ n = \ n = \ {i（i-1）/2+1+1+1 \} $ and $ s_n = 0 $否则，我们将获得nortect of-norm normeger，我们将获得inte norte of-moreger：for-monger：for-momeg for lor-mOnge，以获取所有信息：分区使得$ \ sum_ {i \ geq 1} \ la_ {i（i-1）/2+1} = m $等于$ m $的平面分区数。此外，我们介绍了一个新的分区，块分区，概括了$ k $的分区。从该分区家族中，我们通过安德鲁斯和保勒在$ k $详细的分区中提供了施密特型定理的概括，并与欧拉（Eulerian）多项式建立了链接。
Let $S=(s_n)_{n\geq 1}$ be a sequence with elements in a commutative monoid $(\mathcal{M},+,0)$. In this paper, we provide an explicit formula for $$\sum_{\la} C(\la) q^{\sum_{n\geq 1} \la_n\cdot s_n}$$ where $\la=(\la_1,\ldots)$ run through some subsets of over-partitions, and $C(\la)$ is a certain product of ``colors'' assigned to the parts of $\la$, and $q^s$ is a formal power of $q$ for $s\in M$. This formula allows us not only to retrieve several known Schmidt-type theorems but also to provide new Schmidt-type theorems for non-periodic sequences $S$. For example, when $(M,+,0)=(\mathbb{Z}_{\geq 0},+,0)$, $s_n=1$ if there exists $i\geq 1$ such $n=\{i(i-1)/2+1\}$ and $s_n=0$ otherwise, we obtain the following statement: for all non-negative integer $m$, the number of partitions such that $\sum_{i\geq 1}\la_{i(i-1)/2+1} =m$ is equal to the number of plane partitions of $m$. Furthermore, we introduce a new family of partitions, the block partitions, generalizing the $k$-elongated partitions. From that family of partitions, we provide a generalization of a Schmidt-type theorem due to Andrews and Paule regarding $k$-elongated partitions and establish a link with the Eulerian polynomials.
