Abstract
In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p(n), the number of partitions of n. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions.
Highlights
In 1999, Frank Schmidt [12] proposed the following problem in the American Mathematical Monthly
We shall treat Theorem 1 in great detail to make clear how Partition Analysis is the ideal tool for managing partition questions of this nature
This paper hopefully will spur efforts to find further natural arithmetic/combinatorial objects generated by modular forms
Summary
In 1999, Frank Schmidt [12] proposed the following problem in the American Mathematical Monthly. A number of Schmidt type partitions arising from partitions on various graphs will be seen to have modular forms as generating functions. In [2], we considered “plane partition diamonds”; i.e., partitions whose parts, ai, lie on the following graph, Fig. 1, with each directed edge indicating ≥ Taking such plane partition diamonds of unrestricted length (i.e., n → ∞), and if instead of adding up all the parts, we only add a1 + a4 + a7 + · · · + a3k+1, we will find in Theorem 4 that the generating function is (−q; q)∞ (q; q)3∞. In (7.3) we define the corresponding infinite family of generating functions, (Dk(q))k≥1, counting Schmidt type partitions, and prove a variety of q-series relations and congruences for k = 2 and k = 3.
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