Systematic study of Schmidt-type partitions via weighted words

Let $S$ be a set of positive integers. In this paper, we provide an explicit formula for $$\sum_{\la} C(\la) q^{\sum_{i\in S} \la_i}$$ 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 $\la$. This formula allows us not only to retrieve several known Schmidt-type theorems but also to provide new Schmidt-type theorems for sets $S$ with non-periodic gaps. The example of $S=\{n(n-1)/2+1:n\in 1\}$ leads to the following statement: for all non-negative integer $m$, the number of partitions such that $\sum_{i\in S}\la_i =m$ 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.Comment: 17 pages. 2 figure