Systematic study of Schmidt-type partitions via weighted words
| Autor: | ISAAC KONAN |
|---|---|
| Přispěvatelé: | KONAN, Isaac |
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | HAL |
| Popis: | 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. 18 pages. 2 figures |
| Databáze: | OpenAIRE |
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